Maths-
General
Easy

Question

blank to the power of straight n straight C subscript straight r plus 1 end subscript plus 2 to the power of straight n straight C subscript straight r plus to the power of straight n straight C subscript straight r minus 1 end subscript equals

  1. blank to the power of n plus 1 end exponent C subscript r
  2. blank to the power of n plus 2 end exponent C subscript r
  3. blank to the power of n plus 2 end exponent C subscript r plus 1 end subscript
  4. n plus 2 C subscript r plus 2 end subscript

The correct answer is: blank to the power of n plus 2 end exponent C subscript r plus 1 end subscript


    nC subscript straight r plus 1 end subscript plus to the power of straight n straight C subscript straight r plus to the power of straight n straight C subscript straight r plus to the power of straight n straight C subscript straight r minus 1 end subscript

    Related Questions to study

    General
    maths-

    A person wishes to make up as many different parties of 10 as he can out of 20 friends, each party consisting of the same number In how many of the parties the same man is found ?

    20 minus 1 C subscript 10 minus 1 end subscript

    A person wishes to make up as many different parties of 10 as he can out of 20 friends, each party consisting of the same number In how many of the parties the same man is found ?

    maths-General
    20 minus 1 C subscript 10 minus 1 end subscript
    General
    Maths-

    A father with 6 children takes 3 at a time to a park without taking the same children How often each child goes to the park?

    A father with 6 children takes 3 at a time to a park without taking the same children How often each child goes to the park?

    Maths-General
    General
    chemistry-

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

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

    chemistry-General
    parallel
    General
    chemistry-

    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
    General
    chemistry-

    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
    General
    Maths-

    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

    Maths-General
    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
    General
    Maths-

    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

    Maths-General
    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
    General
    Maths-

    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

    Maths-General
    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
    General
    Maths-

    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

    Maths-General
    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
    General
    Maths-

    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

    Maths-General
    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
    General
    Maths-

    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

    Maths-General
    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
    General
    Maths-

    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

    Maths-General
    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
    General
    Maths-

    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

    Maths-General
    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
    General
    Maths-

    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

    Given, 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

    Maths-General
    Given, 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
    General
    maths-

    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

    maths-General
    parallel

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