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General

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Question

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=

0
$- \frac{1}{3}$
$\frac{2}{3}$
$\frac{- 2}{3}$

Hint:

In the question we will first use L hospital theorem, $fraction numerator f left parenthesis x right parenthesis over denominator g left parenthesis x right parenthesis end fraction equals fraction numerator f apostrophe left parenthesis x right parenthesis over denominator g apostrophe left parenthesis x right parenthesis end fraction$ , in LHS. Then we will use the value of x to get the value of k.

The correct answer is: $2 over 3$

$L straight t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator log begin display style space end style left parenthesis 3 plus x right parenthesis minus log space left parenthesis 3 minus x right parenthesis over denominator x end fraction equals k$
L.H.S= $L straight t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator log begin display style space end style left parenthesis 3 plus x right parenthesis minus log space left parenthesis 3 minus x right parenthesis over denominator x end fraction$
using L hospital theorem,
$L straight t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator begin display style fraction numerator d blank over denominator d straight x end fraction end style log begin display style space end style left parenthesis 3 plus x right parenthesis minus begin display style fraction numerator d blank over denominator d x end fraction end style log space left parenthesis 3 minus x right parenthesis over denominator begin display style fraction numerator d blank over denominator d x end fraction end style x end fraction equals Lt subscript straight x not stretchy rightwards arrow 0 end subscript fraction numerator begin display style fraction numerator 1 over denominator 3 plus straight x end fraction end style plus begin display style fraction numerator 1 over denominator 3 minus straight x end fraction end style over denominator 1 end fraction equals Lt subscript straight x not stretchy rightwards arrow 0 end subscript fraction numerator 3 minus straight x plus 3 plus straight x over denominator left parenthesis 3 plus straight x right parenthesis left parenthesis 3 minus straight x right parenthesis end fraction equals Lt subscript straight x not stretchy rightwards arrow 0 end subscript fraction numerator 6 over denominator 9 plus straight x squared end fraction equals fraction numerator 6 over denominator 9 plus 0 end fraction equals 6 over 9 equals 2 over 3$
So, k= $2 over 3$

Related Questions to study

Let $proportional to$ and $beta$ be the distinct roots of $a x squared plus b x plus c equals 0$ then $fraction numerator L t over denominator x plus a end fraction fraction numerator 1 minus cos space open parentheses a x squared plus b x plus c close parentheses over denominator left parenthesis x minus alpha right parenthesis squared end fraction$

Let $proportional to$ and $beta$ be the distinct roots of $a x squared plus b x plus c equals 0$ then $fraction numerator L t over denominator x plus a end fraction fraction numerator 1 minus cos space open parentheses a x squared plus b x plus c close parentheses over denominator left parenthesis x minus alpha right parenthesis squared end fraction$

In $straight capital delta A B C comma sum a cubed sin invisible function application left parenthesis B minus C right parenthesis equals$

In space straight capital delta A B C comma sum a cubed sin invisible function application left parenthesis B minus C right parenthesis equals space straight a cubed sin left parenthesis straight B minus straight C right parenthesis plus straight b cubed sin left parenthesis straight C minus straight A right parenthesis plus straight c cubed left parenthesis straight A minus straight B right parenthesis L e t space fraction numerator sin space A thin space over denominator a end fraction equals fraction numerator sin space B over denominator b end fraction equals fraction numerator sin space C over denominator c end fraction equals k cos space A equals fraction numerator b squared plus c squared minus a squared over denominator 2 b c end fraction comma cos space A equals fraction numerator c squared plus a squared minus b squared over denominator 2 c a end fraction comma cos space A equals fraction numerator a squared plus b squared minus c squared over denominator 2 a b end fraction space space space space space space straight a cubed sin left parenthesis straight B minus straight C right parenthesis equals a cubed open parentheses fraction numerator k b left parenthesis a squared plus b squared minus c squared right parenthesis over denominator 2 a b end fraction minus fraction numerator k c left parenthesis c squared plus a squared minus b squared right parenthesis over denominator 2 a c end fraction close parentheses equals fraction numerator k a squared over denominator 2 end fraction open parentheses a squared plus b squared minus c squared minus c squared minus a squared plus b squared close parentheses equals fraction numerator k a squared over denominator 2 end fraction open parentheses 2 b squared minus 2 c squared close parentheses equals fraction numerator k a squared over denominator 2 end fraction 2. open parentheses b squared minus c squared close parentheses equals k a squared open parentheses b squared minus c squared close parentheses s i m i l a r l y comma space space b cubed sin left parenthesis C minus A right parenthesis equals k b squared left parenthesis c squared minus a squared right parenthesis s i m i l a r l y comma space c cubed sin space left parenthesis A minus B right parenthesis equals k c squared left parenthesis a squared minus b squared right parenthesis N o w comma space sum a cubed sin invisible function application left parenthesis B minus C right parenthesis equals space straight a cubed sin left parenthesis straight B minus straight C right parenthesis plus straight b cubed sin left parenthesis straight C minus straight A right parenthesis plus straight c cubed left parenthesis straight A minus straight B right parenthesis equals k a squared open parentheses b squared minus c squared close parentheses plus k b squared left parenthesis c squared minus a squared right parenthesis plus k c squared left parenthesis a squared minus b squared right parenthesis equals k a squared b squared minus k a squared c squared plus k b squared c squared minus k a squared b squared plus k a squared c squared minus k b squared c squared equals 0.

In $straight capital delta A B C comma sum a cubed sin invisible function application left parenthesis B minus C right parenthesis equals$

In space straight capital delta A B C comma sum a cubed sin invisible function application left parenthesis B minus C right parenthesis equals space straight a cubed sin left parenthesis straight B minus straight C right parenthesis plus straight b cubed sin left parenthesis straight C minus straight A right parenthesis plus straight c cubed left parenthesis straight A minus straight B right parenthesis L e t space fraction numerator sin space A thin space over denominator a end fraction equals fraction numerator sin space B over denominator b end fraction equals fraction numerator sin space C over denominator c end fraction equals k cos space A equals fraction numerator b squared plus c squared minus a squared over denominator 2 b c end fraction comma cos space A equals fraction numerator c squared plus a squared minus b squared over denominator 2 c a end fraction comma cos space A equals fraction numerator a squared plus b squared minus c squared over denominator 2 a b end fraction space space space space space space straight a cubed sin left parenthesis straight B minus straight C right parenthesis equals a cubed open parentheses fraction numerator k b left parenthesis a squared plus b squared minus c squared right parenthesis over denominator 2 a b end fraction minus fraction numerator k c left parenthesis c squared plus a squared minus b squared right parenthesis over denominator 2 a c end fraction close parentheses equals fraction numerator k a squared over denominator 2 end fraction open parentheses a squared plus b squared minus c squared minus c squared minus a squared plus b squared close parentheses equals fraction numerator k a squared over denominator 2 end fraction open parentheses 2 b squared minus 2 c squared close parentheses equals fraction numerator k a squared over denominator 2 end fraction 2. open parentheses b squared minus c squared close parentheses equals k a squared open parentheses b squared minus c squared close parentheses s i m i l a r l y comma space space b cubed sin left parenthesis C minus A right parenthesis equals k b squared left parenthesis c squared minus a squared right parenthesis s i m i l a r l y comma space c cubed sin space left parenthesis A minus B right parenthesis equals k c squared left parenthesis a squared minus b squared right parenthesis N o w comma space sum a cubed sin invisible function application left parenthesis B minus C right parenthesis equals space straight a cubed sin left parenthesis straight B minus straight C right parenthesis plus straight b cubed sin left parenthesis straight C minus straight A right parenthesis plus straight c cubed left parenthesis straight A minus straight B right parenthesis equals k a squared open parentheses b squared minus c squared close parentheses plus k b squared left parenthesis c squared minus a squared right parenthesis plus k c squared left parenthesis a squared minus b squared right parenthesis equals k a squared b squared minus k a squared c squared plus k b squared c squared minus k a squared b squared plus k a squared c squared minus k b squared c squared equals 0.

$Lt subscript x not stretchy rightwards arrow 1 end subscript space left parenthesis 1 minus x right parenthesis tan space open parentheses pi x over 2 close parentheses equals$

$Lt subscript x not stretchy rightwards arrow 1 end subscript space left parenthesis 1 minus x right parenthesis tan space open parentheses pi x over 2 close parentheses equals$

parallel

In $straight capital delta A B C comma a squared cot invisible function application A plus b squared cot invisible function application B plus C squared cot invisible function application C equals$

increment equals fraction numerator a over denominator 2 sin space A end fraction equals fraction numerator b over denominator 2 space sin space b end fraction equals fraction numerator c over denominator 2 space sin space C end fraction a squared c o t space A plus b squared c o t space B plus c squared c o t space B equals left parenthesis 2 increment sin space A right parenthesis squared. c o t space A plus open parentheses 2 space increment sin space B close parentheses squared. c o t space B plus open parentheses 2 space increment sin space close parentheses squared. c o t space C equals 4 increment squared left parenthesis sin squared space A. c o t space A plus sin squared B. c o t space B plus sin squared C. c o t space C right parenthesis equals 2 increment squared left parenthesis 2 sin space A. cos space A plus 2 sin space B space. cos space B plus 2 sin space C. cos space C right parenthesis equals 2 increment squared left parenthesis sin space 2 A plus sin space 2 B plus sin space 2 C right parenthesis equals 2 increment squared left parenthesis 2 sin left parenthesis A plus B right parenthesis. cos left parenthesis A minus B right parenthesis plus 2. sin space C. cos space C right parenthesis equals 2 increment squared left parenthesis 2 sin left parenthesis straight pi minus straight c right parenthesis. cos left parenthesis A minus B right parenthesis plus 2. sin space C. cos space left parenthesis straight pi minus left parenthesis straight A plus straight B right parenthesis right parenthesis right parenthesis equals 4 increment squared sin space C left parenthesis cos left parenthesis A plus B right parenthesis plus cos left parenthesis A minus B right parenthesis right parenthesis equals 8 increment squared sin space A. sin space B. space sin space C equals 8 increment squared fraction numerator a over denominator 2 increment end fraction. fraction numerator b over denominator 2 increment end fraction. fraction numerator c over denominator 2 increment end fraction equals fraction numerator a b c over denominator increment end fraction

In $straight capital delta A B C comma a squared cot invisible function application A plus b squared cot invisible function application B plus C squared cot invisible function application C equals$

increment equals fraction numerator a over denominator 2 sin space A end fraction equals fraction numerator b over denominator 2 space sin space b end fraction equals fraction numerator c over denominator 2 space sin space C end fraction a squared c o t space A plus b squared c o t space B plus c squared c o t space B equals left parenthesis 2 increment sin space A right parenthesis squared. c o t space A plus open parentheses 2 space increment sin space B close parentheses squared. c o t space B plus open parentheses 2 space increment sin space close parentheses squared. c o t space C equals 4 increment squared left parenthesis sin squared space A. c o t space A plus sin squared B. c o t space B plus sin squared C. c o t space C right parenthesis equals 2 increment squared left parenthesis 2 sin space A. cos space A plus 2 sin space B space. cos space B plus 2 sin space C. cos space C right parenthesis equals 2 increment squared left parenthesis sin space 2 A plus sin space 2 B plus sin space 2 C right parenthesis equals 2 increment squared left parenthesis 2 sin left parenthesis A plus B right parenthesis. cos left parenthesis A minus B right parenthesis plus 2. sin space C. cos space C right parenthesis equals 2 increment squared left parenthesis 2 sin left parenthesis straight pi minus straight c right parenthesis. cos left parenthesis A minus B right parenthesis plus 2. sin space C. cos space left parenthesis straight pi minus left parenthesis straight A plus straight B right parenthesis right parenthesis right parenthesis equals 4 increment squared sin space C left parenthesis cos left parenthesis A plus B right parenthesis plus cos left parenthesis A minus B right parenthesis right parenthesis equals 8 increment squared sin space A. sin space B. space sin space C equals 8 increment squared fraction numerator a over denominator 2 increment end fraction. fraction numerator b over denominator 2 increment end fraction. fraction numerator c over denominator 2 increment end fraction equals fraction numerator a b c over denominator increment end fraction

$Lt subscript x not stretchy rightwards arrow pi over 2 end subscript space fraction numerator open square brackets 1 minus tan space x over 2 close square brackets left square bracket 1 minus sin space x right square bracket over denominator open square brackets 1 plus tan space x over 2 close square brackets left square bracket pi minus 2 x right square bracket cubed end fraction equals$

space space space space Lt subscript x not stretchy rightwards arrow pi over 2 end subscript space fraction numerator open square brackets 1 minus tan space x over 2 close square brackets left square bracket 1 minus sin space x right square bracket over denominator open square brackets 1 plus tan space x over 2 close square brackets left square bracket pi minus 2 x right square bracket cubed end fraction equals Lt subscript x not stretchy rightwards arrow pi over 2 end subscript space fraction numerator open square brackets tan begin display style straight pi over 4 end style minus tan space x over 2 close square brackets over denominator open square brackets tan space begin display style straight pi over 4 end style plus tan space x over 2 close square brackets end fraction. fraction numerator left square bracket 1 minus sin space x right square bracket over denominator left square bracket pi minus 2 x right square bracket cubed end fraction equals Lt subscript x not stretchy rightwards arrow pi over 2 end subscript space tan space open parentheses straight pi over 4 minus x over 2 close parentheses. fraction numerator left square bracket 1 minus sin space x right square bracket over denominator left square bracket pi minus 2 x right square bracket cubed end fraction equals Lt subscript x not stretchy rightwards arrow pi over 2 end subscript space fraction numerator tan space open parentheses begin display style fraction numerator straight pi minus 2 straight x over denominator 4 end fraction end style close parentheses over denominator open parentheses begin display style fraction numerator straight pi minus 2 straight x over denominator 4 end fraction end style close parentheses end fraction. fraction numerator left square bracket 1 minus sin space x right square bracket over denominator 4 left square bracket pi minus 2 x right square bracket squared end fraction equals 1 fourth space Lt subscript x not stretchy rightwards arrow pi over 2 end subscript fraction numerator left square bracket 1 minus sin space x right square bracket over denominator left square bracket pi minus 2 x right square bracket squared end fraction U sin g space L apostrophe H o s p i t a l space t h e o r e m comma space equals 1 fourth space Lt subscript x not stretchy rightwards arrow pi over 2 end subscript fraction numerator negative cos space x over denominator 2 left parenthesis straight pi minus 2 straight x right parenthesis. left parenthesis negative 2 right parenthesis end fraction equals 1 over 16 space Lt subscript x not stretchy rightwards arrow pi over 2 end subscript fraction numerator cos space x over denominator left parenthesis straight pi minus 2 straight x right parenthesis end fraction equals 1 over 16 Lt subscript x not stretchy rightwards arrow pi over 2 end subscript fraction numerator cos space x over denominator straight pi minus 2 straight x end fraction equals 1 over 16 Lt subscript x not stretchy rightwards arrow pi over 2 end subscript fraction numerator cos space x over denominator straight pi minus 2 straight x end fraction U sin g space L apostrophe H o s p i t a l space t h e o r e m comma space equals 1 over 16 Lt subscript x not stretchy rightwards arrow pi over 2 end subscript fraction numerator negative sin space x over denominator negative 2 end fraction equals 1 over 32 Lt subscript x not stretchy rightwards arrow pi over 2 end subscript space sin space x equals 1 over 32 cross times sin space straight pi over 2 equals 1 over 32 space cross times 1 equals 1 over 32.

$Lt subscript x not stretchy rightwards arrow pi over 2 end subscript space fraction numerator open square brackets 1 minus tan space x over 2 close square brackets left square bracket 1 minus sin space x right square bracket over denominator open square brackets 1 plus tan space x over 2 close square brackets left square bracket pi minus 2 x right square bracket cubed end fraction equals$

space space space space Lt subscript x not stretchy rightwards arrow pi over 2 end subscript space fraction numerator open square brackets 1 minus tan space x over 2 close square brackets left square bracket 1 minus sin space x right square bracket over denominator open square brackets 1 plus tan space x over 2 close square brackets left square bracket pi minus 2 x right square bracket cubed end fraction equals Lt subscript x not stretchy rightwards arrow pi over 2 end subscript space fraction numerator open square brackets tan begin display style straight pi over 4 end style minus tan space x over 2 close square brackets over denominator open square brackets tan space begin display style straight pi over 4 end style plus tan space x over 2 close square brackets end fraction. fraction numerator left square bracket 1 minus sin space x right square bracket over denominator left square bracket pi minus 2 x right square bracket cubed end fraction equals Lt subscript x not stretchy rightwards arrow pi over 2 end subscript space tan space open parentheses straight pi over 4 minus x over 2 close parentheses. fraction numerator left square bracket 1 minus sin space x right square bracket over denominator left square bracket pi minus 2 x right square bracket cubed end fraction equals Lt subscript x not stretchy rightwards arrow pi over 2 end subscript space fraction numerator tan space open parentheses begin display style fraction numerator straight pi minus 2 straight x over denominator 4 end fraction end style close parentheses over denominator open parentheses begin display style fraction numerator straight pi minus 2 straight x over denominator 4 end fraction end style close parentheses end fraction. fraction numerator left square bracket 1 minus sin space x right square bracket over denominator 4 left square bracket pi minus 2 x right square bracket squared end fraction equals 1 fourth space Lt subscript x not stretchy rightwards arrow pi over 2 end subscript fraction numerator left square bracket 1 minus sin space x right square bracket over denominator left square bracket pi minus 2 x right square bracket squared end fraction U sin g space L apostrophe H o s p i t a l space t h e o r e m comma space equals 1 fourth space Lt subscript x not stretchy rightwards arrow pi over 2 end subscript fraction numerator negative cos space x over denominator 2 left parenthesis straight pi minus 2 straight x right parenthesis. left parenthesis negative 2 right parenthesis end fraction equals 1 over 16 space Lt subscript x not stretchy rightwards arrow pi over 2 end subscript fraction numerator cos space x over denominator left parenthesis straight pi minus 2 straight x right parenthesis end fraction equals 1 over 16 Lt subscript x not stretchy rightwards arrow pi over 2 end subscript fraction numerator cos space x over denominator straight pi minus 2 straight x end fraction equals 1 over 16 Lt subscript x not stretchy rightwards arrow pi over 2 end subscript fraction numerator cos space x over denominator straight pi minus 2 straight x end fraction U sin g space L apostrophe H o s p i t a l space t h e o r e m comma space equals 1 over 16 Lt subscript x not stretchy rightwards arrow pi over 2 end subscript fraction numerator negative sin space x over denominator negative 2 end fraction equals 1 over 32 Lt subscript x not stretchy rightwards arrow pi over 2 end subscript space sin space x equals 1 over 32 cross times sin space straight pi over 2 equals 1 over 32 space cross times 1 equals 1 over 32.

$Lt subscript x not stretchy rightwards arrow a end subscript fraction numerator square root of a plus 2 x end root minus square root of 3 x end root over denominator square root of 3 a plus x end root minus 2 square root of x end fraction equals$

space space space Lt subscript x not stretchy rightwards arrow a end subscript fraction numerator square root of a plus 2 x end root minus square root of 3 x end root over denominator square root of 3 a plus x end root minus 2 square root of x end fraction R a t o n a l i sin g space t h e space n u m e r a t o r comma equals Lt subscript x not stretchy rightwards arrow a end subscript fraction numerator square root of a plus 2 x end root minus square root of 3 x end root over denominator square root of 3 a plus x end root minus 2 square root of x end fraction cross times fraction numerator square root of a plus 2 x end root plus square root of 3 x end root over denominator square root of a plus 2 x end root plus square root of 3 x end root end fraction cross times fraction numerator square root of 3 a plus x end root plus 2 square root of x over denominator square root of 3 a plus x end root plus 2 square root of x end fraction equals Lt subscript x not stretchy rightwards arrow a end subscript fraction numerator left parenthesis a plus 2 x minus 3 x right parenthesis open parentheses square root of 3 a plus x end root plus 2 square root of x close parentheses over denominator left parenthesis 3 a plus x minus 4 x right parenthesis open parentheses square root of a plus 2 x end root plus square root of 3 x end root close parentheses end fraction equals Lt subscript x not stretchy rightwards arrow a end subscript fraction numerator square root of 3 a plus x end root plus 2 square root of x over denominator 3 open parentheses square root of a plus 2 x end root plus square root of 3 x end root close parentheses end fraction equals fraction numerator square root of 3 a plus a end root plus 2 square root of a over denominator 3 open parentheses square root of a plus 2 a end root plus square root of 3 a end root close parentheses end fraction equals 1 third cross times fraction numerator 4 square root of a over denominator 2 square root of 3 a end root end fraction equals fraction numerator 2 over denominator 3 square root of 3 end fraction

$Lt subscript x not stretchy rightwards arrow a end subscript fraction numerator square root of a plus 2 x end root minus square root of 3 x end root over denominator square root of 3 a plus x end root minus 2 square root of x end fraction equals$

space space space Lt subscript x not stretchy rightwards arrow a end subscript fraction numerator square root of a plus 2 x end root minus square root of 3 x end root over denominator square root of 3 a plus x end root minus 2 square root of x end fraction R a t o n a l i sin g space t h e space n u m e r a t o r comma equals Lt subscript x not stretchy rightwards arrow a end subscript fraction numerator square root of a plus 2 x end root minus square root of 3 x end root over denominator square root of 3 a plus x end root minus 2 square root of x end fraction cross times fraction numerator square root of a plus 2 x end root plus square root of 3 x end root over denominator square root of a plus 2 x end root plus square root of 3 x end root end fraction cross times fraction numerator square root of 3 a plus x end root plus 2 square root of x over denominator square root of 3 a plus x end root plus 2 square root of x end fraction equals Lt subscript x not stretchy rightwards arrow a end subscript fraction numerator left parenthesis a plus 2 x minus 3 x right parenthesis open parentheses square root of 3 a plus x end root plus 2 square root of x close parentheses over denominator left parenthesis 3 a plus x minus 4 x right parenthesis open parentheses square root of a plus 2 x end root plus square root of 3 x end root close parentheses end fraction equals Lt subscript x not stretchy rightwards arrow a end subscript fraction numerator square root of 3 a plus x end root plus 2 square root of x over denominator 3 open parentheses square root of a plus 2 x end root plus square root of 3 x end root close parentheses end fraction equals fraction numerator square root of 3 a plus a end root plus 2 square root of a over denominator 3 open parentheses square root of a plus 2 a end root plus square root of 3 a end root close parentheses end fraction equals 1 third cross times fraction numerator 4 square root of a over denominator 2 square root of 3 a end root end fraction equals fraction numerator 2 over denominator 3 square root of 3 end fraction

parallel

In $straight triangle A B C comma cot invisible function application A over 2 plus cot invisible function application B over 2 plus cot invisible function application C over 2 equals$

space space space c o t space A over 2 plus c o t space B over 2 plus c o t space C over 2 equals square root of fraction numerator s left parenthesis s minus a right parenthesis over denominator left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis end fraction end root plus square root of fraction numerator s left parenthesis s minus b right parenthesis over denominator left parenthesis s minus a right parenthesis left parenthesis s minus c right parenthesis end fraction end root plus square root of fraction numerator s left parenthesis s minus c right parenthesis over denominator left parenthesis s minus a right parenthesis left parenthesis s minus b right parenthesis end fraction end root equals square root of fraction numerator s over denominator left parenthesis s minus a right parenthesis left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis end fraction end root left curly bracket left parenthesis s minus a right parenthesis plus left parenthesis s minus b right parenthesis plus left parenthesis s minus c right parenthesis right curly bracket m u l t i p l y i n g space t h e space n u m e r a t o r space a n d space d e n o m i n a t o r space w i t h space square root of s equals fraction numerator s over denominator square root of s left parenthesis s minus a right parenthesis left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis end root end fraction left curly bracket 3 s minus left parenthesis a plus b plus c right parenthesis right curly bracket equals fraction numerator s left parenthesis 3 s minus left parenthesis a plus b plus c right parenthesis right parenthesis over denominator increment end fraction equals fraction numerator begin display style fraction numerator left parenthesis a plus b plus c right parenthesis over denominator 2 end fraction end style open square brackets begin display style fraction numerator 3 open parentheses a plus b plus c close parentheses over denominator 2 end fraction end style minus begin display style fraction numerator 2 open parentheses a plus b plus c close parentheses over denominator 2 end fraction end style close square brackets over denominator increment end fraction equals fraction numerator open parentheses a plus b plus c close parentheses squared over denominator 4 increment end fraction

In $straight triangle A B C comma cot invisible function application A over 2 plus cot invisible function application B over 2 plus cot invisible function application C over 2 equals$

space space space c o t space A over 2 plus c o t space B over 2 plus c o t space C over 2 equals square root of fraction numerator s left parenthesis s minus a right parenthesis over denominator left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis end fraction end root plus square root of fraction numerator s left parenthesis s minus b right parenthesis over denominator left parenthesis s minus a right parenthesis left parenthesis s minus c right parenthesis end fraction end root plus square root of fraction numerator s left parenthesis s minus c right parenthesis over denominator left parenthesis s minus a right parenthesis left parenthesis s minus b right parenthesis end fraction end root equals square root of fraction numerator s over denominator left parenthesis s minus a right parenthesis left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis end fraction end root left curly bracket left parenthesis s minus a right parenthesis plus left parenthesis s minus b right parenthesis plus left parenthesis s minus c right parenthesis right curly bracket m u l t i p l y i n g space t h e space n u m e r a t o r space a n d space d e n o m i n a t o r space w i t h space square root of s equals fraction numerator s over denominator square root of s left parenthesis s minus a right parenthesis left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis end root end fraction left curly bracket 3 s minus left parenthesis a plus b plus c right parenthesis right curly bracket equals fraction numerator s left parenthesis 3 s minus left parenthesis a plus b plus c right parenthesis right parenthesis over denominator increment end fraction equals fraction numerator begin display style fraction numerator left parenthesis a plus b plus c right parenthesis over denominator 2 end fraction end style open square brackets begin display style fraction numerator 3 open parentheses a plus b plus c close parentheses over denominator 2 end fraction end style minus begin display style fraction numerator 2 open parentheses a plus b plus c close parentheses over denominator 2 end fraction end style close square brackets over denominator increment end fraction equals fraction numerator open parentheses a plus b plus c close parentheses squared over denominator 4 increment end fraction

$Lt subscript x not stretchy rightwards arrow 1 end subscript fraction numerator left parenthesis 2 x minus 3 right parenthesis left parenthesis square root of x minus 1 right parenthesis over denominator 2 x squared plus x minus 3 end fraction equals$

Lt subscript x not stretchy rightwards arrow 1 end subscript fraction numerator left parenthesis 2 x minus 3 right parenthesis left parenthesis square root of x minus 1 right parenthesis over denominator 2 x squared plus x minus 3 end fraction equals Lt subscript x not stretchy rightwards arrow 1 end subscript fraction numerator 2 x to the power of begin display style 3 over 2 end style end exponent minus 3 to the power of begin display style 1 half end style end exponent minus 2 x plus 3 over denominator 2 x squared plus x minus 3 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 1 end subscript fraction numerator begin display style fraction numerator d blank over denominator d x end fraction end style 2 x to the power of begin display style 3 over 2 end style end exponent minus 3 to the power of begin display style 1 half end style end exponent minus 2 x plus 3 over denominator begin display style fraction numerator d blank over denominator d x end fraction end style 2 x squared plus x minus 3 end fraction equals Lt subscript x not stretchy rightwards arrow 1 end subscript fraction numerator 3 square root of x minus begin display style fraction numerator 3 over denominator 2 square root of x end fraction end style minus 2 over denominator 4 x plus 1 end fraction equals fraction numerator 3 cross times 1 space minus begin display style 3 over 2 end style cross times 1 minus 2 over denominator 4 cross times 1 space plus 1 end fraction equals fraction numerator 3 minus begin display style 3 over 2 end style minus 2 over denominator 5 end fraction equals fraction numerator begin display style fraction numerator 6 minus 3 minus 4 over denominator 2 end fraction end style over denominator 5 end fraction equals fraction numerator negative 1 over denominator 10 end fraction

$Lt subscript x not stretchy rightwards arrow 1 end subscript fraction numerator left parenthesis 2 x minus 3 right parenthesis left parenthesis square root of x minus 1 right parenthesis over denominator 2 x squared plus x minus 3 end fraction equals$

Lt subscript x not stretchy rightwards arrow 1 end subscript fraction numerator left parenthesis 2 x minus 3 right parenthesis left parenthesis square root of x minus 1 right parenthesis over denominator 2 x squared plus x minus 3 end fraction equals Lt subscript x not stretchy rightwards arrow 1 end subscript fraction numerator 2 x to the power of begin display style 3 over 2 end style end exponent minus 3 to the power of begin display style 1 half end style end exponent minus 2 x plus 3 over denominator 2 x squared plus x minus 3 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 1 end subscript fraction numerator begin display style fraction numerator d blank over denominator d x end fraction end style 2 x to the power of begin display style 3 over 2 end style end exponent minus 3 to the power of begin display style 1 half end style end exponent minus 2 x plus 3 over denominator begin display style fraction numerator d blank over denominator d x end fraction end style 2 x squared plus x minus 3 end fraction equals Lt subscript x not stretchy rightwards arrow 1 end subscript fraction numerator 3 square root of x minus begin display style fraction numerator 3 over denominator 2 square root of x end fraction end style minus 2 over denominator 4 x plus 1 end fraction equals fraction numerator 3 cross times 1 space minus begin display style 3 over 2 end style cross times 1 minus 2 over denominator 4 cross times 1 space plus 1 end fraction equals fraction numerator 3 minus begin display style 3 over 2 end style minus 2 over denominator 5 end fraction equals fraction numerator begin display style fraction numerator 6 minus 3 minus 4 over denominator 2 end fraction end style over denominator 5 end fraction equals fraction numerator negative 1 over denominator 10 end fraction

$Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 11 x cubed minus 3 x plus 4 over denominator 13 x cubed minus 5 x squared minus 7 end fraction$

Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 11 x cubed minus 3 x plus 4 over denominator 13 x cubed minus 5 x squared minus 7 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 straight infinity end subscript fraction numerator begin display style fraction numerator d over denominator d x end fraction end style 11 x cubed minus 3 x plus 4 over denominator begin display style fraction numerator d over denominator d x end fraction end style 13 x cubed minus 5 x squared minus 7 end fraction equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 33 x squared minus 3 over denominator 39 x squared minus 10 x end fraction A g a i n comma space 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 straight infinity end subscript fraction numerator begin display style fraction numerator d blank over denominator d x end fraction end style 33 x squared minus 3 over denominator begin display style fraction numerator d blank over denominator d x end fraction end style 39 x squared minus 10 x end fraction equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 66 x over denominator 78 x minus 10 end fraction A g a i n comma space 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 straight infinity end subscript fraction numerator begin display style fraction numerator d blank over denominator d x end fraction end style 66 x over denominator begin display style fraction numerator d blank over denominator d x end fraction end style 78 x minus 10 end fraction equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript 66 over 78 equals 66 over 78 equals 11 over 13

$Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 11 x cubed minus 3 x plus 4 over denominator 13 x cubed minus 5 x squared minus 7 end fraction$

Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 11 x cubed minus 3 x plus 4 over denominator 13 x cubed minus 5 x squared minus 7 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 straight infinity end subscript fraction numerator begin display style fraction numerator d over denominator d x end fraction end style 11 x cubed minus 3 x plus 4 over denominator begin display style fraction numerator d over denominator d x end fraction end style 13 x cubed minus 5 x squared minus 7 end fraction equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 33 x squared minus 3 over denominator 39 x squared minus 10 x end fraction A g a i n comma space 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 straight infinity end subscript fraction numerator begin display style fraction numerator d blank over denominator d x end fraction end style 33 x squared minus 3 over denominator begin display style fraction numerator d blank over denominator d x end fraction end style 39 x squared minus 10 x end fraction equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 66 x over denominator 78 x minus 10 end fraction A g a i n comma space 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 straight infinity end subscript fraction numerator begin display style fraction numerator d blank over denominator d x end fraction end style 66 x over denominator begin display style fraction numerator d blank over denominator d x end fraction end style 78 x minus 10 end fraction equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript 66 over 78 equals 66 over 78 equals 11 over 13

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In $straight capital delta A B C$ , cot $A plus cot invisible function application B plus cot invisible function application C equals$

space space space space c o t space A plus c o t space B plus space c o t space C space equals fraction numerator cos space A over denominator sin A end fraction plus fraction numerator cos space B space over denominator sin space B space end fraction plus fraction numerator cos space C over denominator sin space C end fraction equals fraction numerator b squared plus c squared minus a squared over denominator 2 b c end fraction. fraction numerator b c over denominator 2 increment end fraction plus fraction numerator c squared plus a squared minus b squared over denominator 2 c a end fraction. fraction numerator c a over denominator 2 increment end fraction plus fraction numerator a squared plus b squared minus c squared over denominator 2 a b end fraction. fraction numerator a b over denominator 2 increment end fraction space space space space space open square brackets u sin g space cos i n e space f o r m u l a space a n d space a r e a space o f space t r i a n g l e space open parentheses increment equals fraction numerator a b space sin C over denominator 2 end fraction equals fraction numerator b c space sin space A over denominator 2 end fraction equals fraction numerator c a space sin space B over denominator 2 end fraction close parentheses close square brackets equals fraction numerator b squared plus c squared minus a squared plus c squared plus a squared minus b squared plus a squared plus b squared minus c squared over denominator 4 increment end fraction equals fraction numerator a squared plus b squared plus c squared over denominator 4 increment end fraction

In $straight capital delta A B C$ , cot $A plus cot invisible function application B plus cot invisible function application C equals$

space space space space c o t space A plus c o t space B plus space c o t space C space equals fraction numerator cos space A over denominator sin A end fraction plus fraction numerator cos space B space over denominator sin space B space end fraction plus fraction numerator cos space C over denominator sin space C end fraction equals fraction numerator b squared plus c squared minus a squared over denominator 2 b c end fraction. fraction numerator b c over denominator 2 increment end fraction plus fraction numerator c squared plus a squared minus b squared over denominator 2 c a end fraction. fraction numerator c a over denominator 2 increment end fraction plus fraction numerator a squared plus b squared minus c squared over denominator 2 a b end fraction. fraction numerator a b over denominator 2 increment end fraction space space space space space open square brackets u sin g space cos i n e space f o r m u l a space a n d space a r e a space o f space t r i a n g l e space open parentheses increment equals fraction numerator a b space sin C over denominator 2 end fraction equals fraction numerator b c space sin space A over denominator 2 end fraction equals fraction numerator c a space sin space B over denominator 2 end fraction close parentheses close square brackets equals fraction numerator b squared plus c squared minus a squared plus c squared plus a squared minus b squared plus a squared plus b squared minus c squared over denominator 4 increment end fraction equals fraction numerator a squared plus b squared plus c squared over denominator 4 increment end fraction

$Lt subscript x not stretchy rightwards arrow straight infinity end subscript space fraction numerator 8 vertical line x vertical line plus 3 x over denominator 3 vertical line x vertical line minus 2 x end fraction$

Lt subscript x not stretchy rightwards arrow straight infinity end subscript space fraction numerator 8 vertical line x vertical line plus 3 x over denominator 3 vertical line x vertical line minus 2 x end fraction

open vertical bar x close vertical bar equals open curly brackets table attributes columnalign left end attributes row cell x space space space comma space x greater or equal than 0 end cell row cell negative x comma space x less than 0 end cell end table close t a k i n g space x space p o s i t i v e comma limit as x rightwards arrow infinity of fraction numerator 8 x plus 3 x over denominator 3 x minus 2 x end fraction equals limit as x rightwards arrow infinity of fraction numerator 11 x over denominator x end fraction equals 11

$Lt subscript x not stretchy rightwards arrow straight infinity end subscript space fraction numerator 8 vertical line x vertical line plus 3 x over denominator 3 vertical line x vertical line minus 2 x end fraction$

Lt subscript x not stretchy rightwards arrow straight infinity end subscript space fraction numerator 8 vertical line x vertical line plus 3 x over denominator 3 vertical line x vertical line minus 2 x end fraction

open vertical bar x close vertical bar equals open curly brackets table attributes columnalign left end attributes row cell x space space space comma space x greater or equal than 0 end cell row cell negative x comma space x less than 0 end cell end table close t a k i n g space x space p o s i t i v e comma limit as x rightwards arrow infinity of fraction numerator 8 x plus 3 x over denominator 3 x minus 2 x end fraction equals limit as x rightwards arrow infinity of fraction numerator 11 x over denominator x end fraction equals 11

$Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 2 plus cos squared space x over denominator x plus 2007 end fraction$

Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 2 plus cos squared space x over denominator x plus 2007 end fraction U sin g space L space H o s p t a l space t h e o r e m comma space equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator begin display style fraction numerator d blank over denominator d x end fraction end style 2 plus cos squared space x over denominator begin display style fraction numerator d blank over denominator d x end fraction end style x plus 2007 end fraction equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 2 space cos space x space sin space x over denominator 1 end fraction equals 2

$Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 2 plus cos squared space x over denominator x plus 2007 end fraction$

Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 2 plus cos squared space x over denominator x plus 2007 end fraction U sin g space L space H o s p t a l space t h e o r e m comma space equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator begin display style fraction numerator d blank over denominator d x end fraction end style 2 plus cos squared space x over denominator begin display style fraction numerator d blank over denominator d x end fraction end style x plus 2007 end fraction equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 2 space cos space x space sin space x over denominator 1 end fraction equals 2

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$Lt subscript x not stretchy rightwards arrow straight infinity end subscript open parentheses square root of x squared plus x end root minus x close parentheses$

space space space space Lt subscript x not stretchy rightwards arrow straight infinity end subscript open parentheses square root of x squared plus x end root minus x close parentheses equals space Lt subscript x not stretchy rightwards arrow straight infinity end subscript open parentheses square root of x squared plus x end root minus x close parentheses cross times fraction numerator square root of x squared plus x end root plus x over denominator square root of x squared plus x end root plus x end fraction equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator x squared plus x minus x squared over denominator square root of x squared plus x end root plus x end fraction equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator x over denominator square root of x squared plus x end root plus x end fraction D i v i d i n g space b y space x comma space equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 1 over denominator square root of 1 plus begin display style 1 over x end style end root plus 1 end fraction equals fraction numerator 1 over denominator square root of 1 plus 1 end fraction equals 1 half.

$Lt subscript x not stretchy rightwards arrow straight infinity end subscript open parentheses square root of x squared plus x end root minus x close parentheses$

space space space space Lt subscript x not stretchy rightwards arrow straight infinity end subscript open parentheses square root of x squared plus x end root minus x close parentheses equals space Lt subscript x not stretchy rightwards arrow straight infinity end subscript open parentheses square root of x squared plus x end root minus x close parentheses cross times fraction numerator square root of x squared plus x end root plus x over denominator square root of x squared plus x end root plus x end fraction equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator x squared plus x minus x squared over denominator square root of x squared plus x end root plus x end fraction equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator x over denominator square root of x squared plus x end root plus x end fraction D i v i d i n g space b y space x comma space equals Lt subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 1 over denominator square root of 1 plus begin display style 1 over x end style end root plus 1 end fraction equals fraction numerator 1 over denominator square root of 1 plus 1 end fraction equals 1 half.

Elastic potential and gravitational potential energies are equal all the time.

The main difference between gravitational potential energy and elastic potential energy is that the origin of gravitational potential energy is the gravitational forces acting between two massive bodies whereas the origin of elastic potential energy is the electrostatic forces between molecules that make up a material
An object possesses gravitational potential energy if it is positioned at a height above (or below) the zero height. An object possesses elastic potential energy if it is at a position on an elastic medium other than the equilibrium position

Elastic potential and gravitational potential energies are equal all the time.

The main difference between gravitational potential energy and elastic potential energy is that the origin of gravitational potential energy is the gravitational forces acting between two massive bodies whereas the origin of elastic potential energy is the electrostatic forces between molecules that make up a material
An object possesses gravitational potential energy if it is positioned at a height above (or below) the zero height. An object possesses elastic potential energy if it is at a position on an elastic medium other than the equilibrium position

In $straight triangle A B C comma$ cot $A plus cot invisible function application B plus cot invisible function application C equals$

space space space space c o t space A plus c o t space B plus space c o t space C space equals fraction numerator cos space A over denominator sin A end fraction plus fraction numerator cos space B space over denominator sin space B space end fraction plus fraction numerator cos space C over denominator sin space C end fraction equals fraction numerator b squared plus c squared minus a squared over denominator 2 b c end fraction. fraction numerator b c over denominator 2 increment end fraction plus fraction numerator c squared plus a squared minus b squared over denominator 2 c a end fraction. fraction numerator c a over denominator 2 increment end fraction plus fraction numerator a squared plus b squared minus c squared over denominator 2 a b end fraction. fraction numerator a b over denominator 2 increment end fraction space space space space space open square brackets u sin g space cos i n e space f o r m u l a space a n d space a r e a space o f space t r i a n g l e space open parentheses increment equals fraction numerator a b space sin C over denominator 2 end fraction equals fraction numerator b c space sin space A over denominator 2 end fraction equals fraction numerator c a space sin space B over denominator 2 end fraction close parentheses close square brackets equals fraction numerator b squared plus c squared minus a squared plus c squared plus a squared minus b squared plus a squared plus b squared minus c squared over denominator 4 increment end fraction equals fraction numerator a squared plus b squared plus c squared over denominator 4 increment end fraction

In $straight triangle A B C comma$ cot $A plus cot invisible function application B plus cot invisible function application C equals$

space space space space c o t space A plus c o t space B plus space c o t space C space equals fraction numerator cos space A over denominator sin A end fraction plus fraction numerator cos space B space over denominator sin space B space end fraction plus fraction numerator cos space C over denominator sin space C end fraction equals fraction numerator b squared plus c squared minus a squared over denominator 2 b c end fraction. fraction numerator b c over denominator 2 increment end fraction plus fraction numerator c squared plus a squared minus b squared over denominator 2 c a end fraction. fraction numerator c a over denominator 2 increment end fraction plus fraction numerator a squared plus b squared minus c squared over denominator 2 a b end fraction. fraction numerator a b over denominator 2 increment end fraction space space space space space open square brackets u sin g space cos i n e space f o r m u l a space a n d space a r e a space o f space t r i a n g l e space open parentheses increment equals fraction numerator a b space sin C over denominator 2 end fraction equals fraction numerator b c space sin space A over denominator 2 end fraction equals fraction numerator c a space sin space B over denominator 2 end fraction close parentheses close square brackets equals fraction numerator b squared plus c squared minus a squared plus c squared plus a squared minus b squared plus a squared plus b squared minus c squared over denominator 4 increment end fraction equals fraction numerator a squared plus b squared plus c squared over denominator 4 increment end fraction

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