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Chemistry-

General

Easy

Question

Action of $H N O subscript 2 space o n space C H subscript 3 N H subscript 2$ gives:

$C H_{3} O H$
$C H_{3} ∙ O ∙ C H_{3}$
$C H_{3} O — N = O$
Both (b) and (c)

The correct answer is: Both (b) and (c)

Related Questions to study

Primary alcohols can be obtained from the reaction of the $R M g X$ with:

Primary alcohols can be obtained from the reaction of the $R M g X$ with:

chemistry-General

$R long dash C H equals C H subscript 2$ reacts with in presence of $H subscript 2 O subscript 2$ to give:

$R long dash C H equals C H subscript 2$ reacts with in presence of $H subscript 2 O subscript 2$ to give:

chemistry-General

The quadratic equation whose roots are I and m where l= $lim for theta not stretchy rightwards arrow 0 of open parentheses fraction numerator 3 sin space theta minus 4 sin cubed space theta over denominator theta end fraction close parentheses m equals lim for theta not stretchy rightwards arrow 0 of open parentheses fraction numerator 2 tan space theta over denominator theta open parentheses 1 minus tan squared space theta close parentheses end fraction close parentheses$ is

Given, the quadratic equation whose roots are I and m where
l =

lim for theta not stretchy rightwards arrow 0 of   open parentheses fraction numerator 3 sin space theta minus 4 sin cubed space theta over denominator theta end fraction close parentheses m equals lim for theta not stretchy rightwards arrow 0 of   open parentheses fraction numerator 2 tan space theta over denominator theta open parentheses 1 minus tan squared space theta close parentheses end fraction close parentheses

l equals limit as theta rightwards arrow 0 of open parentheses fraction numerator 3 space sin theta minus 4 sin cubed theta over denominator theta end fraction close parentheses space space space space space space space space space space space space space space space space space space space space m equals limit as theta rightwards arrow 0 of open parentheses fraction numerator 2 tan theta over denominator theta left parenthesis 1 minus tan squared theta right parenthesis end fraction close parentheses space equals limit as theta rightwards arrow 0 of open parentheses fraction numerator sin space 3 theta over denominator theta end fraction close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals limit as theta rightwards arrow 0 of open parentheses fraction numerator tan space 2 theta over denominator theta end fraction close parentheses space equals limit as theta rightwards arrow 0 of open parentheses fraction numerator sin space 3 theta over denominator 3 theta end fraction close parentheses cross times 3 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals limit as theta rightwards arrow 0 of space open parentheses fraction numerator sin 2 theta over denominator theta cos 2 theta end fraction close parentheses space equals 1 cross times 3 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals limit as theta rightwards arrow 0 of open parentheses fraction numerator sin 2 theta over denominator 2 theta end fraction close parentheses cross times 2 cross times fraction numerator 1 over denominator cos 2 theta end fraction space equals 3 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 2 l equals 3 space a n d space m equals 2 t h e space q u a d t r a t i c space e q u a t i o n space w h e n space t h e space r o o t s space a r e space g i v e n equals x squared minus left parenthesis l plus m right parenthesis x plus l m equals x squared minus left parenthesis 3 plus 2 right parenthesis x plus left parenthesis 3 cross times 2 right parenthesis equals x squared minus 5 x plus 6 space

The quadratic equation whose roots are I and m where l= $lim for theta not stretchy rightwards arrow 0 of open parentheses fraction numerator 3 sin space theta minus 4 sin cubed space theta over denominator theta end fraction close parentheses m equals lim for theta not stretchy rightwards arrow 0 of open parentheses fraction numerator 2 tan space theta over denominator theta open parentheses 1 minus tan squared space theta close parentheses end fraction close parentheses$ is

Given, the quadratic equation whose roots are I and m where
l =

lim for theta not stretchy rightwards arrow 0 of   open parentheses fraction numerator 3 sin space theta minus 4 sin cubed space theta over denominator theta end fraction close parentheses m equals lim for theta not stretchy rightwards arrow 0 of   open parentheses fraction numerator 2 tan space theta over denominator theta open parentheses 1 minus tan squared space theta close parentheses end fraction close parentheses

l equals limit as theta rightwards arrow 0 of open parentheses fraction numerator 3 space sin theta minus 4 sin cubed theta over denominator theta end fraction close parentheses space space space space space space space space space space space space space space space space space space space space m equals limit as theta rightwards arrow 0 of open parentheses fraction numerator 2 tan theta over denominator theta left parenthesis 1 minus tan squared theta right parenthesis end fraction close parentheses space equals limit as theta rightwards arrow 0 of open parentheses fraction numerator sin space 3 theta over denominator theta end fraction close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals limit as theta rightwards arrow 0 of open parentheses fraction numerator tan space 2 theta over denominator theta end fraction close parentheses space equals limit as theta rightwards arrow 0 of open parentheses fraction numerator sin space 3 theta over denominator 3 theta end fraction close parentheses cross times 3 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals limit as theta rightwards arrow 0 of space open parentheses fraction numerator sin 2 theta over denominator theta cos 2 theta end fraction close parentheses space equals 1 cross times 3 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals limit as theta rightwards arrow 0 of open parentheses fraction numerator sin 2 theta over denominator 2 theta end fraction close parentheses cross times 2 cross times fraction numerator 1 over denominator cos 2 theta end fraction space equals 3 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 2 l equals 3 space a n d space m equals 2 t h e space q u a d t r a t i c space e q u a t i o n space w h e n space t h e space r o o t s space a r e space g i v e n equals x squared minus left parenthesis l plus m right parenthesis x plus l m equals x squared minus left parenthesis 3 plus 2 right parenthesis x plus left parenthesis 3 cross times 2 right parenthesis equals x squared minus 5 x plus 6 space

parallel

$L subscript x not stretchy rightwards arrow 0 end subscript equals fraction numerator tan cubed begin display style space end style x minus sin cubed space x over denominator x to the power of 5 end fraction$

L subscript x not stretchy rightwards arrow 0 end subscript fraction numerator tan cubed begin display style space end style x minus sin cubed space x over denominator x to the power of 5 end fraction equals limit as x rightwards arrow 0 of fraction numerator tan cubed x minus begin display style fraction numerator sin squared x. space cos squared x over denominator cos squared x end fraction end style over denominator x to the power of 5 end fraction equals limit as x rightwards arrow 0 of fraction numerator tan cubed x left parenthesis 1 minus begin display style space cos squared x end style right parenthesis over denominator x to the power of 5 end fraction equals limit as x rightwards arrow 0 of open parentheses fraction numerator tan cubed x over denominator x cubed end fraction close parentheses limit as x rightwards arrow 0 of open parentheses fraction numerator 1 minus cos cubed x over denominator x squared end fraction close parentheses limit as x rightwards arrow 0 of open parentheses fraction numerator 1 minus cos cubed x over denominator x squared end fraction close parentheses u sin g space L space H o s p i t a l space t h e o r e m comma limit as x rightwards arrow 0 of open parentheses fraction numerator begin display style fraction numerator d blank over denominator d x end fraction end style left parenthesis 1 minus cos cubed x right parenthesis over denominator begin display style fraction numerator d blank over denominator d x end fraction end style x squared end fraction close parentheses equals limit as x rightwards arrow 0 of fraction numerator 3 cos squared x. sin x over denominator 2 x end fraction equals limit as x rightwards arrow 0 of 3 over 2 cos squared x. open parentheses fraction numerator sin x over denominator x end fraction close parentheses equals 3 over 2 limit as x rightwards arrow 0 of cos squared x equals 3 over 2 cos squared 0 equals 3 over 2 N o w comma limit as x rightwards arrow 0 of open parentheses fraction numerator tan cubed x over denominator x cubed end fraction close parentheses limit as x rightwards arrow 0 of open parentheses fraction numerator 1 minus cos cubed x over denominator x squared end fraction close parentheses equals 1 cross times 3 over 2 equals 3 over 2

$L subscript x not stretchy rightwards arrow 0 end subscript equals fraction numerator tan cubed begin display style space end style x minus sin cubed space x over denominator x to the power of 5 end fraction$

L subscript x not stretchy rightwards arrow 0 end subscript fraction numerator tan cubed begin display style space end style x minus sin cubed space x over denominator x to the power of 5 end fraction equals limit as x rightwards arrow 0 of fraction numerator tan cubed x minus begin display style fraction numerator sin squared x. space cos squared x over denominator cos squared x end fraction end style over denominator x to the power of 5 end fraction equals limit as x rightwards arrow 0 of fraction numerator tan cubed x left parenthesis 1 minus begin display style space cos squared x end style right parenthesis over denominator x to the power of 5 end fraction equals limit as x rightwards arrow 0 of open parentheses fraction numerator tan cubed x over denominator x cubed end fraction close parentheses limit as x rightwards arrow 0 of open parentheses fraction numerator 1 minus cos cubed x over denominator x squared end fraction close parentheses limit as x rightwards arrow 0 of open parentheses fraction numerator 1 minus cos cubed x over denominator x squared end fraction close parentheses u sin g space L space H o s p i t a l space t h e o r e m comma limit as x rightwards arrow 0 of open parentheses fraction numerator begin display style fraction numerator d blank over denominator d x end fraction end style left parenthesis 1 minus cos cubed x right parenthesis over denominator begin display style fraction numerator d blank over denominator d x end fraction end style x squared end fraction close parentheses equals limit as x rightwards arrow 0 of fraction numerator 3 cos squared x. sin x over denominator 2 x end fraction equals limit as x rightwards arrow 0 of 3 over 2 cos squared x. open parentheses fraction numerator sin x over denominator x end fraction close parentheses equals 3 over 2 limit as x rightwards arrow 0 of cos squared x equals 3 over 2 cos squared 0 equals 3 over 2 N o w comma limit as x rightwards arrow 0 of open parentheses fraction numerator tan cubed x over denominator x cubed end fraction close parentheses limit as x rightwards arrow 0 of open parentheses fraction numerator 1 minus cos cubed x over denominator x squared end fraction close parentheses equals 1 cross times 3 over 2 equals 3 over 2

$L subscript x not stretchy rightwards arrow 0 end subscript equals fraction numerator sin begin display style space end style open parentheses pi cos squared space x close parentheses over denominator x squared end fraction$ Is equal to

L subscript x not stretchy rightwards arrow 0 end subscript fraction numerator sin begin display style space end style open parentheses pi cos squared space x close parentheses over denominator x squared end fraction equals L subscript x not stretchy rightwards arrow 0 end subscript fraction numerator sin space open parentheses straight pi open parentheses 1 minus sin squared straight x close parentheses close parentheses over denominator x squared end fraction equals straight pi open curly brackets straight L subscript straight x not stretchy rightwards arrow 0 end subscript fraction numerator sin open parentheses straight pi minus πsin squared straight x close parentheses over denominator πsin squared straight x end fraction cross times fraction numerator sin squared straight x over denominator straight x squared end fraction close curly brackets space space space space space space space space space open square brackets sin left parenthesis straight pi minus straight a right parenthesis equals sina close square brackets equals straight pi open curly brackets straight L subscript straight x not stretchy rightwards arrow 0 end subscript fraction numerator sin open parentheses πsin squared straight x close parentheses over denominator πsin squared straight x end fraction cross times fraction numerator sin squared straight x over denominator straight x squared end fraction close curly brackets space space space space space space space space space space space space equals straight pi cross times 1 cross times 1 equals straight pi

$L subscript x not stretchy rightwards arrow 0 end subscript equals fraction numerator sin begin display style space end style open parentheses pi cos squared space x close parentheses over denominator x squared end fraction$ Is equal to

L subscript x not stretchy rightwards arrow 0 end subscript fraction numerator sin begin display style space end style open parentheses pi cos squared space x close parentheses over denominator x squared end fraction equals L subscript x not stretchy rightwards arrow 0 end subscript fraction numerator sin space open parentheses straight pi open parentheses 1 minus sin squared straight x close parentheses close parentheses over denominator x squared end fraction equals straight pi open curly brackets straight L subscript straight x not stretchy rightwards arrow 0 end subscript fraction numerator sin open parentheses straight pi minus πsin squared straight x close parentheses over denominator πsin squared straight x end fraction cross times fraction numerator sin squared straight x over denominator straight x squared end fraction close curly brackets space space space space space space space space space open square brackets sin left parenthesis straight pi minus straight a right parenthesis equals sina close square brackets equals straight pi open curly brackets straight L subscript straight x not stretchy rightwards arrow 0 end subscript fraction numerator sin open parentheses πsin squared straight x close parentheses over denominator πsin squared straight x end fraction cross times fraction numerator sin squared straight x over denominator straight x squared end fraction close curly brackets space space space space space space space space space space space space equals straight pi cross times 1 cross times 1 equals straight pi

$L t subscript x plus 0 end subscript fraction numerator left parenthesis 1 minus cos space 2 x right parenthesis left parenthesis 3 plus cos space x right parenthesis over denominator x tan space 4 x end fraction$ is equal to

L t subscript x rightwards arrow 0 end subscript fraction numerator left parenthesis 1 minus cos space 2 x right parenthesis left parenthesis 3 plus cos space x right parenthesis over denominator x tan space 4 x end fraction equals L t subscript x rightwards arrow 0 end subscript space fraction numerator open parentheses 1 minus 1 plus 2 sin squared x close parentheses open parentheses 3 plus cos x close parentheses over denominator x.4 x end fraction fraction numerator 4 x over denominator tan 4 x end fraction space space space space space space space space space space space space space space space space space space space open square brackets cos space 2 x equals 1 minus 2 sin squared x close square brackets equals L t subscript x rightwards arrow 0 end subscript fraction numerator 2 sin squared x over denominator 4 x squared end fraction space L t subscript x rightwards arrow 0 end subscript open parentheses 3 plus cos x close parentheses L t subscript x rightwards arrow 0 end subscript fraction numerator 4 x over denominator tan 4 x end fraction equals 1 half L t subscript x rightwards arrow 0 end subscript fraction numerator sin squared x over denominator x squared end fraction space L t subscript x rightwards arrow 0 end subscript open parentheses 3 plus cos x close parentheses L t subscript x rightwards arrow 0 end subscript fraction numerator 4 x over denominator tan 4 x end fraction equals 1 half open parentheses 3 plus 1 close parentheses 1 equals 4 over 2 equals 2

$L t subscript x plus 0 end subscript fraction numerator left parenthesis 1 minus cos space 2 x right parenthesis left parenthesis 3 plus cos space x right parenthesis over denominator x tan space 4 x end fraction$ is equal to

L t subscript x rightwards arrow 0 end subscript fraction numerator left parenthesis 1 minus cos space 2 x right parenthesis left parenthesis 3 plus cos space x right parenthesis over denominator x tan space 4 x end fraction equals L t subscript x rightwards arrow 0 end subscript space fraction numerator open parentheses 1 minus 1 plus 2 sin squared x close parentheses open parentheses 3 plus cos x close parentheses over denominator x.4 x end fraction fraction numerator 4 x over denominator tan 4 x end fraction space space space space space space space space space space space space space space space space space space space open square brackets cos space 2 x equals 1 minus 2 sin squared x close square brackets equals L t subscript x rightwards arrow 0 end subscript fraction numerator 2 sin squared x over denominator 4 x squared end fraction space L t subscript x rightwards arrow 0 end subscript open parentheses 3 plus cos x close parentheses L t subscript x rightwards arrow 0 end subscript fraction numerator 4 x over denominator tan 4 x end fraction equals 1 half L t subscript x rightwards arrow 0 end subscript fraction numerator sin squared x over denominator x squared end fraction space L t subscript x rightwards arrow 0 end subscript open parentheses 3 plus cos x close parentheses L t subscript x rightwards arrow 0 end subscript fraction numerator 4 x over denominator tan 4 x end fraction equals 1 half open parentheses 3 plus 1 close parentheses 1 equals 4 over 2 equals 2

parallel

$Lt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator tan begin display style space end style x minus sin space x over denominator x squared end fraction equals$

Lt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator tan begin display style space end style x minus sin space x over denominator x squared end fraction equals limit as x rightwards arrow 0 of fraction numerator begin display style fraction numerator sin x over denominator cos x end fraction end style minus sin x over denominator x squared end fraction equals limit as x rightwards arrow 0 of fraction numerator begin display style fraction numerator sin x minus sin x space cos x over denominator cos x end fraction end style over denominator x squared end fraction equals limit as x rightwards arrow 0 of fraction numerator sin x left parenthesis 1 minus cos x right parenthesis over denominator x squared. cos x end fraction equals limit as x rightwards arrow 0 of fraction numerator sin x space over denominator x end fraction fraction numerator x over denominator cos x end fraction fraction numerator 1 minus cos x over denominator x squared end fraction equals limit as x rightwards arrow 0 of fraction numerator sin space x over denominator x end fraction limit as x rightwards arrow 0 of fraction numerator x over denominator cos space x end fraction limit as x rightwards arrow 0 of fraction numerator 2 sin squared begin display style x over 2 end style over denominator open parentheses begin display style x over 2 end style close parentheses squared 4 end fraction space space space space space space space space space space space space space space space space space space open square brackets cos space 2 x equals 1 minus 2 sin squared x close square brackets equals 1. fraction numerator 0 over denominator cos 0 end fraction.2 equals 0

$Lt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator tan begin display style space end style x minus sin space x over denominator x squared end fraction equals$

Lt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator tan begin display style space end style x minus sin space x over denominator x squared end fraction equals limit as x rightwards arrow 0 of fraction numerator begin display style fraction numerator sin x over denominator cos x end fraction end style minus sin x over denominator x squared end fraction equals limit as x rightwards arrow 0 of fraction numerator begin display style fraction numerator sin x minus sin x space cos x over denominator cos x end fraction end style over denominator x squared end fraction equals limit as x rightwards arrow 0 of fraction numerator sin x left parenthesis 1 minus cos x right parenthesis over denominator x squared. cos x end fraction equals limit as x rightwards arrow 0 of fraction numerator sin x space over denominator x end fraction fraction numerator x over denominator cos x end fraction fraction numerator 1 minus cos x over denominator x squared end fraction equals limit as x rightwards arrow 0 of fraction numerator sin space x over denominator x end fraction limit as x rightwards arrow 0 of fraction numerator x over denominator cos space x end fraction limit as x rightwards arrow 0 of fraction numerator 2 sin squared begin display style x over 2 end style over denominator open parentheses begin display style x over 2 end style close parentheses squared 4 end fraction space space space space space space space space space space space space space space space space space space open square brackets cos space 2 x equals 1 minus 2 sin squared x close square brackets equals 1. fraction numerator 0 over denominator cos 0 end fraction.2 equals 0

$Lt subscript x not stretchy rightwards arrow 8 end subscript space fraction numerator square root of 1 plus square root of 1 plus x end root end root minus 2 over denominator x minus 8 end fraction equals$

Lt subscript x not stretchy rightwards arrow 8 end subscript space fraction numerator square root of 1 plus square root of 1 plus x end root end root minus 2 over denominator x minus 8 end fraction L e t space y equals square root of 1 plus x end root space space space space space space space space space space space space space space space space space space space space x rightwards arrow 8 rightwards double arrow y squared equals 1 plus x space space space space space space space space space space space space space space space space space space space space space space space y rightwards arrow square root of 1 plus x end root rightwards double arrow x equals y squared minus 1 space space space space space space space space space space space space space space space space space space space space space space space y rightwards arrow square root of 1 plus 8 end root space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space y rightwards arrow 3 Lt subscript x not stretchy rightwards arrow 8 end subscript space fraction numerator square root of 1 plus square root of 1 plus x end root end root minus 2 over denominator x minus 8 end fraction equals limit as y rightwards arrow 3 of fraction numerator square root of 1 plus y end root minus 2 over denominator y squared minus 1 minus 8 end fraction equals limit as y rightwards arrow 3 of fraction numerator square root of 1 plus y end root minus 2 over denominator y squared minus 9 end fraction equals limit as y rightwards arrow 3 of fraction numerator open parentheses square root of 1 plus y end root minus 2 close parentheses open parentheses square root of 1 plus y end root plus 2 close parentheses over denominator open parentheses y squared minus 9 close parentheses open parentheses square root of 1 plus y end root plus 2 close parentheses end fraction equals limit as y rightwards arrow 3 of fraction numerator 1 plus y minus 4 over denominator open parentheses y squared minus 9 close parentheses open parentheses square root of 1 plus y end root plus 2 close parentheses end fraction equals limit as y rightwards arrow 3 of fraction numerator y minus 3 over denominator open parentheses y squared minus 3 squared close parentheses open parentheses square root of 1 plus y end root plus 2 close parentheses end fraction equals limit as y rightwards arrow 3 of fraction numerator y minus 3 over denominator left parenthesis y minus 3 right parenthesis left parenthesis y plus 3 right parenthesis open parentheses square root of 1 plus y end root plus 2 close parentheses end fraction equals limit as y rightwards arrow 3 of fraction numerator 1 over denominator open parentheses y plus 3 close parentheses open parentheses square root of 1 plus y end root plus 2 close parentheses end fraction equals fraction numerator 1 over denominator left parenthesis 3 plus 3 right parenthesis left parenthesis square root of 1 plus 3 end root plus 2 right parenthesis end fraction equals fraction numerator 1 over denominator 6 cross times 4 end fraction equals 1 over 24

$Lt subscript x not stretchy rightwards arrow 8 end subscript space fraction numerator square root of 1 plus square root of 1 plus x end root end root minus 2 over denominator x minus 8 end fraction equals$

Lt subscript x not stretchy rightwards arrow 8 end subscript space fraction numerator square root of 1 plus square root of 1 plus x end root end root minus 2 over denominator x minus 8 end fraction L e t space y equals square root of 1 plus x end root space space space space space space space space space space space space space space space space space space space space x rightwards arrow 8 rightwards double arrow y squared equals 1 plus x space space space space space space space space space space space space space space space space space space space space space space space y rightwards arrow square root of 1 plus x end root rightwards double arrow x equals y squared minus 1 space space space space space space space space space space space space space space space space space space space space space space space y rightwards arrow square root of 1 plus 8 end root space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space y rightwards arrow 3 Lt subscript x not stretchy rightwards arrow 8 end subscript space fraction numerator square root of 1 plus square root of 1 plus x end root end root minus 2 over denominator x minus 8 end fraction equals limit as y rightwards arrow 3 of fraction numerator square root of 1 plus y end root minus 2 over denominator y squared minus 1 minus 8 end fraction equals limit as y rightwards arrow 3 of fraction numerator square root of 1 plus y end root minus 2 over denominator y squared minus 9 end fraction equals limit as y rightwards arrow 3 of fraction numerator open parentheses square root of 1 plus y end root minus 2 close parentheses open parentheses square root of 1 plus y end root plus 2 close parentheses over denominator open parentheses y squared minus 9 close parentheses open parentheses square root of 1 plus y end root plus 2 close parentheses end fraction equals limit as y rightwards arrow 3 of fraction numerator 1 plus y minus 4 over denominator open parentheses y squared minus 9 close parentheses open parentheses square root of 1 plus y end root plus 2 close parentheses end fraction equals limit as y rightwards arrow 3 of fraction numerator y minus 3 over denominator open parentheses y squared minus 3 squared close parentheses open parentheses square root of 1 plus y end root plus 2 close parentheses end fraction equals limit as y rightwards arrow 3 of fraction numerator y minus 3 over denominator left parenthesis y minus 3 right parenthesis left parenthesis y plus 3 right parenthesis open parentheses square root of 1 plus y end root plus 2 close parentheses end fraction equals limit as y rightwards arrow 3 of fraction numerator 1 over denominator open parentheses y plus 3 close parentheses open parentheses square root of 1 plus y end root plus 2 close parentheses end fraction equals fraction numerator 1 over denominator left parenthesis 3 plus 3 right parenthesis left parenthesis square root of 1 plus 3 end root plus 2 right parenthesis end fraction equals fraction numerator 1 over denominator 6 cross times 4 end fraction equals 1 over 24

$L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator x over denominator square root of x plus 4 end root minus 2 end fraction equals$

L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator x over denominator square root of x plus 4 end root minus 2 end fraction u sin g space L h o s p i t a l space t h e o r e m comma space equals L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator begin display style fraction numerator d blank over denominator d x end fraction end style x over denominator begin display style fraction numerator d blank over denominator d x end fraction end style square root of x plus 4 end root minus 2 end fraction equals L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator 1 over denominator begin display style fraction numerator 1 over denominator 2 square root of x plus 4 end root end fraction end style end fraction equals L t subscript x not stretchy rightwards arrow 0 end subscript 2 square root of x plus 4 end root equals 2 square root of 0 plus 4 end root equals 2 cross times 2 equals 4

$L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator x over denominator square root of x plus 4 end root minus 2 end fraction equals$

L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator x over denominator square root of x plus 4 end root minus 2 end fraction u sin g space L h o s p i t a l space t h e o r e m comma space equals L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator begin display style fraction numerator d blank over denominator d x end fraction end style x over denominator begin display style fraction numerator d blank over denominator d x end fraction end style square root of x plus 4 end root minus 2 end fraction equals L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator 1 over denominator begin display style fraction numerator 1 over denominator 2 square root of x plus 4 end root end fraction end style end fraction equals L t subscript x not stretchy rightwards arrow 0 end subscript 2 square root of x plus 4 end root equals 2 square root of 0 plus 4 end root equals 2 cross times 2 equals 4

parallel

$Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator cube root of 1 plus x end root minus cube root of 1 minus x end root over denominator x end fraction equals$

Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator cube root of 1 plus x end root minus cube root of 1 minus x end root over denominator x end fraction u sin g space L space h o s p i t a l space t h e o r e m comma space equals Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator begin display style fraction numerator d blank over denominator d x end fraction end style cube root of 1 plus x end root minus begin display style fraction numerator d blank over denominator d x end fraction end style cube root of 1 minus x end root over denominator begin display style fraction numerator d blank over denominator d x end fraction end style x end fraction equals Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator begin display style fraction numerator 1 over denominator 3 open parentheses 1 plus x close parentheses to the power of begin display style 2 over 3 end style end exponent end fraction end style minus begin display style fraction numerator negative 1 over denominator 3 open parentheses 1 minus x close parentheses to the power of begin display style 2 over 3 end style end exponent end fraction end style over denominator 1 end fraction equals Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator 1 over denominator 3 open parentheses 1 plus x close parentheses to the power of begin display style 2 over 3 end style end exponent end fraction plus fraction numerator 1 over denominator 3 open parentheses 1 minus x close parentheses to the power of begin display style 2 over 3 end style end exponent end fraction equals fraction numerator 1 over denominator 3 open parentheses 1 plus 0 close parentheses to the power of begin display style 2 over 3 end style end exponent end fraction plus fraction numerator 1 over denominator 3 open parentheses 1 minus 0 close parentheses to the power of begin display style 2 over 3 end style end exponent end fraction equals 1 third plus 1 third equals 2 over 3

$Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator cube root of 1 plus x end root minus cube root of 1 minus x end root over denominator x end fraction equals$

Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator cube root of 1 plus x end root minus cube root of 1 minus x end root over denominator x end fraction u sin g space L space h o s p i t a l space t h e o r e m comma space equals Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator begin display style fraction numerator d blank over denominator d x end fraction end style cube root of 1 plus x end root minus begin display style fraction numerator d blank over denominator d x end fraction end style cube root of 1 minus x end root over denominator begin display style fraction numerator d blank over denominator d x end fraction end style x end fraction equals Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator begin display style fraction numerator 1 over denominator 3 open parentheses 1 plus x close parentheses to the power of begin display style 2 over 3 end style end exponent end fraction end style minus begin display style fraction numerator negative 1 over denominator 3 open parentheses 1 minus x close parentheses to the power of begin display style 2 over 3 end style end exponent end fraction end style over denominator 1 end fraction equals Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator 1 over denominator 3 open parentheses 1 plus x close parentheses to the power of begin display style 2 over 3 end style end exponent end fraction plus fraction numerator 1 over denominator 3 open parentheses 1 minus x close parentheses to the power of begin display style 2 over 3 end style end exponent end fraction equals fraction numerator 1 over denominator 3 open parentheses 1 plus 0 close parentheses to the power of begin display style 2 over 3 end style end exponent end fraction plus fraction numerator 1 over denominator 3 open parentheses 1 minus 0 close parentheses to the power of begin display style 2 over 3 end style end exponent end fraction equals 1 third plus 1 third equals 2 over 3

If $f left parenthesis x right parenthesis equals negative square root of 25 minus x squared end root comma text then end text L t subscript x not stretchy rightwards arrow 1 end subscript fraction numerator f left parenthesis x right parenthesis minus f left parenthesis 1 right parenthesis over denominator x minus 1 end fraction equals$

f left parenthesis x right parenthesis equals negative square root of 25 minus x squared end root

L t subscript x not stretchy rightwards arrow 1 end subscript fraction numerator f left parenthesis x right parenthesis minus f left parenthesis 1 right parenthesis over denominator x minus 1 end fraction u sin g space L space h o s p i t a l space t h e o r e m comma equals L t subscript x not stretchy rightwards arrow 1 end subscript fraction numerator begin display style fraction numerator d blank over denominator d x end fraction end style f left parenthesis x right parenthesis minus f left parenthesis 1 right parenthesis over denominator begin display style fraction numerator d blank over denominator d x end fraction end style x minus 1 end fraction equals L t subscript x not stretchy rightwards arrow 1 end subscript fraction numerator f apostrophe left parenthesis x right parenthesis over denominator 1 end fraction equals L t subscript x not stretchy rightwards arrow 1 space end subscript f apostrophe left parenthesis x right parenthesis equals L t subscript x not stretchy rightwards arrow 1 space end subscript fraction numerator d left parenthesis negative square root of 25 minus x squared end root right parenthesis over denominator d x end fraction equals L t subscript x not stretchy rightwards arrow 1 space end subscript fraction numerator negative 1 cross times negative 2 over denominator 2 square root of 25 minus x squared end root end fraction equals L t subscript x not stretchy rightwards arrow 1 space end subscript fraction numerator 1 over denominator square root of 25 minus x squared end root end fraction equals fraction numerator 1 over denominator square root of 25 minus 1 end root end fraction equals fraction numerator 1 over denominator square root of 24 end fraction

If $f left parenthesis x right parenthesis equals negative square root of 25 minus x squared end root comma text then end text L t subscript x not stretchy rightwards arrow 1 end subscript fraction numerator f left parenthesis x right parenthesis minus f left parenthesis 1 right parenthesis over denominator x minus 1 end fraction equals$

f left parenthesis x right parenthesis equals negative square root of 25 minus x squared end root

L t subscript x not stretchy rightwards arrow 1 end subscript fraction numerator f left parenthesis x right parenthesis minus f left parenthesis 1 right parenthesis over denominator x minus 1 end fraction u sin g space L space h o s p i t a l space t h e o r e m comma equals L t subscript x not stretchy rightwards arrow 1 end subscript fraction numerator begin display style fraction numerator d blank over denominator d x end fraction end style f left parenthesis x right parenthesis minus f left parenthesis 1 right parenthesis over denominator begin display style fraction numerator d blank over denominator d x end fraction end style x minus 1 end fraction equals L t subscript x not stretchy rightwards arrow 1 end subscript fraction numerator f apostrophe left parenthesis x right parenthesis over denominator 1 end fraction equals L t subscript x not stretchy rightwards arrow 1 space end subscript f apostrophe left parenthesis x right parenthesis equals L t subscript x not stretchy rightwards arrow 1 space end subscript fraction numerator d left parenthesis negative square root of 25 minus x squared end root right parenthesis over denominator d x end fraction equals L t subscript x not stretchy rightwards arrow 1 space end subscript fraction numerator negative 1 cross times negative 2 over denominator 2 square root of 25 minus x squared end root end fraction equals L t subscript x not stretchy rightwards arrow 1 space end subscript fraction numerator 1 over denominator square root of 25 minus x squared end root end fraction equals fraction numerator 1 over denominator square root of 25 minus 1 end root end fraction equals fraction numerator 1 over denominator square root of 24 end fraction

$L t subscript x not stretchy rightwards arrow 2 end subscript open parentheses fraction numerator square root of 1 minus cos space left curly bracket 2 left parenthesis x minus 2 right parenthesis factorial end root over denominator x minus 2 end fraction close parentheses equals$

$L t subscript x not stretchy rightwards arrow 2 end subscript open parentheses fraction numerator square root of 1 minus cos space left curly bracket 2 left parenthesis x minus 2 right parenthesis factorial end root over denominator x minus 2 end fraction close parentheses equals$

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If $L t subscript x not stretchy rightwards arrow straight infinity end subscript open parentheses 1 plus a over x plus b over x squared close parentheses to the power of 2 x end exponent equals e squared$ then the values of a and b are

space space space space L t subscript x not stretchy rightwards arrow straight infinity end subscript open parentheses 1 plus a over x plus b over x squared close parentheses to the power of 2 x end exponent equals e squared rightwards double arrow e to the power of limit as x rightwards arrow infinity of 2 x open parentheses 1 plus a over x plus b over x squared minus 1 close parentheses end exponent equals e to the power of 2 space space space space space space space space space end exponent space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets u sin g space f left parenthesis x right parenthesis to the power of g left parenthesis x right parenthesis end exponent equals g left parenthesis x right parenthesis open square brackets f left parenthesis x right parenthesis minus 1 close square brackets close square brackets rightwards double arrow limit as x rightwards arrow infinity of 2 x open parentheses a over x plus b over x squared close parentheses equals 2 rightwards double arrow 2 limit as x rightwards arrow infinity of open parentheses a plus b over x close parentheses equals 2 rightwards double arrow 2. left parenthesis a plus 0 right parenthesis equals 2 rightwards double arrow a equals 1 space a n d space b element of R

If $L t subscript x not stretchy rightwards arrow straight infinity end subscript open parentheses 1 plus a over x plus b over x squared close parentheses to the power of 2 x end exponent equals e squared$ then the values of a and b are

space space space space L t subscript x not stretchy rightwards arrow straight infinity end subscript open parentheses 1 plus a over x plus b over x squared close parentheses to the power of 2 x end exponent equals e squared rightwards double arrow e to the power of limit as x rightwards arrow infinity of 2 x open parentheses 1 plus a over x plus b over x squared minus 1 close parentheses end exponent equals e to the power of 2 space space space space space space space space space end exponent space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets u sin g space f left parenthesis x right parenthesis to the power of g left parenthesis x right parenthesis end exponent equals g left parenthesis x right parenthesis open square brackets f left parenthesis x right parenthesis minus 1 close square brackets close square brackets rightwards double arrow limit as x rightwards arrow infinity of 2 x open parentheses a over x plus b over x squared close parentheses equals 2 rightwards double arrow 2 limit as x rightwards arrow infinity of open parentheses a plus b over x close parentheses equals 2 rightwards double arrow 2. left parenthesis a plus 0 right parenthesis equals 2 rightwards double arrow a equals 1 space a n d space b element of R

$text In end text straight triangle A B C comma straight capital sigma fraction numerator a squared sin invisible function application left parenthesis B minus C right parenthesis over denominator sin invisible function application A end fraction equals$

text  In  end text straight triangle A B C comma straight capital sigma fraction numerator a squared sin degree left parenthesis B minus C right parenthesis over denominator sin degree A end fraction equals fraction numerator straight a squared sin degree left parenthesis straight B minus straight C right parenthesis over denominator sin degree straight A end fraction plus fraction numerator straight b squared sin degree left parenthesis straight C minus straight A right parenthesis over denominator sin degree straight B end fraction plus fraction numerator straight c squared sin degree left parenthesis straight A minus straight B right parenthesis over denominator sin degree straight C end fraction using space sine space formula comma fraction numerator straight a over denominator sin space straight A end fraction equals fraction numerator straight b over denominator sin space straight B end fraction equals fraction numerator straight c over denominator sin space straight C end fraction equals straight k fraction numerator straight a squared sin degree left parenthesis straight B minus straight C right parenthesis over denominator sin degree straight A end fraction plus fraction numerator straight b squared sin degree left parenthesis straight C minus straight A right parenthesis over denominator sin degree straight B end fraction plus fraction numerator straight c squared sin degree left parenthesis straight A minus straight B right parenthesis over denominator sin degree straight C end fraction equals fraction numerator straight k squared sin squared straight A open parentheses sin open parentheses straight B minus straight C close parentheses close parentheses over denominator sinA end fraction plus fraction numerator straight k squared sin squared straight B open parentheses sin open parentheses straight C minus straight A close parentheses close parentheses over denominator sin space straight B end fraction plus fraction numerator straight k squared sin squared straight C open parentheses sin open parentheses straight A minus straight B close parentheses close parentheses over denominator sinC end fraction equals straight k squared open square brackets sin space straight A space sin open parentheses straight B minus straight C close parentheses plus sinB left parenthesis sin open parentheses straight C minus straight A close parentheses plus sinC open parentheses sin open parentheses straight A minus straight B close parentheses close parentheses close square brackets equals straight k squared open square brackets sin open parentheses straight pi minus open parentheses straight B plus straight C close parentheses close parentheses space sin open parentheses straight B minus straight C close parentheses plus sin open parentheses straight pi minus open parentheses straight C plus straight A close parentheses close parentheses left parenthesis sin open parentheses straight C minus straight A close parentheses plus sin open parentheses straight pi minus open parentheses straight A plus straight B close parentheses close parentheses open parentheses sin open parentheses straight A minus straight B close parentheses close parentheses close square brackets equals straight k squared open square brackets sin left parenthesis straight B plus straight C right parenthesis space sin open parentheses straight B minus straight C close parentheses plus sin left parenthesis straight C plus straight A right parenthesis left parenthesis sin open parentheses straight C minus straight A close parentheses plus sin left parenthesis straight A plus straight B right parenthesis open parentheses sin open parentheses straight A minus straight B close parentheses close parentheses close square brackets equals straight k squared open square brackets sin squared straight B minus sin squared straight C plus sin squared straight C minus sin squared straight A plus sin squared straight A minus sin squared straight B close square brackets equals straight k squared cross times 0 equals 0.

$text In end text straight triangle A B C comma straight capital sigma fraction numerator a squared sin invisible function application left parenthesis B minus C right parenthesis over denominator sin invisible function application A end fraction equals$

text  In  end text straight triangle A B C comma straight capital sigma fraction numerator a squared sin degree left parenthesis B minus C right parenthesis over denominator sin degree A end fraction equals fraction numerator straight a squared sin degree left parenthesis straight B minus straight C right parenthesis over denominator sin degree straight A end fraction plus fraction numerator straight b squared sin degree left parenthesis straight C minus straight A right parenthesis over denominator sin degree straight B end fraction plus fraction numerator straight c squared sin degree left parenthesis straight A minus straight B right parenthesis over denominator sin degree straight C end fraction using space sine space formula comma fraction numerator straight a over denominator sin space straight A end fraction equals fraction numerator straight b over denominator sin space straight B end fraction equals fraction numerator straight c over denominator sin space straight C end fraction equals straight k fraction numerator straight a squared sin degree left parenthesis straight B minus straight C right parenthesis over denominator sin degree straight A end fraction plus fraction numerator straight b squared sin degree left parenthesis straight C minus straight A right parenthesis over denominator sin degree straight B end fraction plus fraction numerator straight c squared sin degree left parenthesis straight A minus straight B right parenthesis over denominator sin degree straight C end fraction equals fraction numerator straight k squared sin squared straight A open parentheses sin open parentheses straight B minus straight C close parentheses close parentheses over denominator sinA end fraction plus fraction numerator straight k squared sin squared straight B open parentheses sin open parentheses straight C minus straight A close parentheses close parentheses over denominator sin space straight B end fraction plus fraction numerator straight k squared sin squared straight C open parentheses sin open parentheses straight A minus straight B close parentheses close parentheses over denominator sinC end fraction equals straight k squared open square brackets sin space straight A space sin open parentheses straight B minus straight C close parentheses plus sinB left parenthesis sin open parentheses straight C minus straight A close parentheses plus sinC open parentheses sin open parentheses straight A minus straight B close parentheses close parentheses close square brackets equals straight k squared open square brackets sin open parentheses straight pi minus open parentheses straight B plus straight C close parentheses close parentheses space sin open parentheses straight B minus straight C close parentheses plus sin open parentheses straight pi minus open parentheses straight C plus straight A close parentheses close parentheses left parenthesis sin open parentheses straight C minus straight A close parentheses plus sin open parentheses straight pi minus open parentheses straight A plus straight B close parentheses close parentheses open parentheses sin open parentheses straight A minus straight B close parentheses close parentheses close square brackets equals straight k squared open square brackets sin left parenthesis straight B plus straight C right parenthesis space sin open parentheses straight B minus straight C close parentheses plus sin left parenthesis straight C plus straight A right parenthesis left parenthesis sin open parentheses straight C minus straight A close parentheses plus sin left parenthesis straight A plus straight B right parenthesis open parentheses sin open parentheses straight A minus straight B close parentheses close parentheses close square brackets equals straight k squared open square brackets sin squared straight B minus sin squared straight C plus sin squared straight C minus sin squared straight A plus sin squared straight A minus sin squared straight B close square brackets equals straight k squared cross times 0 equals 0.

If $L straight t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator log begin display style space end style left parenthesis 3 plus x right parenthesis minus log space left parenthesis 3 minus x right parenthesis over denominator x end fraction equals k$ then k=

L straight t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator log begin display style space end style left parenthesis 3 plus x right parenthesis minus log space left parenthesis 3 minus x right parenthesis over denominator x end fraction equals k

L straight t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator log begin display style space end style left parenthesis 3 plus x right parenthesis minus log space left parenthesis 3 minus x right parenthesis over denominator x end fraction

using L hospital theorem,

L straight t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator begin display style fraction numerator d blank over denominator d straight x end fraction end style log begin display style space end style left parenthesis 3 plus x right parenthesis minus begin display style fraction numerator d blank over denominator d x end fraction end style log space left parenthesis 3 minus x right parenthesis over denominator begin display style fraction numerator d blank over denominator d x end fraction end style x end fraction equals Lt subscript straight x not stretchy rightwards arrow 0 end subscript fraction numerator begin display style fraction numerator 1 over denominator 3 plus straight x end fraction end style plus begin display style fraction numerator 1 over denominator 3 minus straight x end fraction end style over denominator 1 end fraction equals Lt subscript straight x not stretchy rightwards arrow 0 end subscript fraction numerator 3 minus straight x plus 3 plus straight x over denominator left parenthesis 3 plus straight x right parenthesis left parenthesis 3 minus straight x right parenthesis end fraction equals Lt subscript straight x not stretchy rightwards arrow 0 end subscript fraction numerator 6 over denominator 9 plus straight x squared end fraction equals fraction numerator 6 over denominator 9 plus 0 end fraction equals 6 over 9 equals 2 over 3

2 over 3

If $L straight t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator log begin display style space end style left parenthesis 3 plus x right parenthesis minus log space left parenthesis 3 minus x right parenthesis over denominator x end fraction equals k$ then k=

L straight t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator log begin display style space end style left parenthesis 3 plus x right parenthesis minus log space left parenthesis 3 minus x right parenthesis over denominator x end fraction equals k

L straight t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator log begin display style space end style left parenthesis 3 plus x right parenthesis minus log space left parenthesis 3 minus x right parenthesis over denominator x end fraction

using L hospital theorem,

L straight t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator begin display style fraction numerator d blank over denominator d straight x end fraction end style log begin display style space end style left parenthesis 3 plus x right parenthesis minus begin display style fraction numerator d blank over denominator d x end fraction end style log space left parenthesis 3 minus x right parenthesis over denominator begin display style fraction numerator d blank over denominator d x end fraction end style x end fraction equals Lt subscript straight x not stretchy rightwards arrow 0 end subscript fraction numerator begin display style fraction numerator 1 over denominator 3 plus straight x end fraction end style plus begin display style fraction numerator 1 over denominator 3 minus straight x end fraction end style over denominator 1 end fraction equals Lt subscript straight x not stretchy rightwards arrow 0 end subscript fraction numerator 3 minus straight x plus 3 plus straight x over denominator left parenthesis 3 plus straight x right parenthesis left parenthesis 3 minus straight x right parenthesis end fraction equals Lt subscript straight x not stretchy rightwards arrow 0 end subscript fraction numerator 6 over denominator 9 plus straight x squared end fraction equals fraction numerator 6 over denominator 9 plus 0 end fraction equals 6 over 9 equals 2 over 3

2 over 3

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