Math, asked by khushi32632, 11 months ago

The value of Lim x—a cosx-cosa/cotx-Cota

Answers

Answered by AJAYMAHICH
3

Step-by-step explanation:

Lim cosx-cosa/cotx-cota

0/0 form so use D-L hospitalization rule :

= Lim -sinx/-cosec^2x

= Lim -sina/-cosec^2 a

= sin^3 a

Answered by Sharad001
61

Question:-

 \sf \: find \: the \: value \: of \: \:   \red{\lim_{x \to a} } \bigg \{\green{ \frac{ \cos x -  \cos a}{ \cot  x -  \cot a} }  \bigg \}\\

Answer :-

\boxed{ \red{\lim_{x \to a} } \bigg \{\green{ \frac{  \blue{\cos x -  \cos a}}{ \cot  x -  \cot a} }  \bigg \} \:  \:  =  { \sin}^{3} a} \:  \:

Solution :-

We have ,

 \implies \pink{ \lim_{x \to a}} \bigg \{ \blue{ \frac{ \cos x -  \cos a}{ \cot  x -  \cot a}  }\bigg \} \\   \\ \bf \: Taking  \: limit  \:  \\  \\  \implies \sf \frac{ \cos a -  \cos a}{ \cot a -  \cot a}  \\  \\  \implies \:  \frac{0}{0}  \:  \{ \sf \green{ indeterminate \: form }\} \\  \\  \bf \: hence \:  \\  \bf \red{Apply  \: L } \: - Hospital  \: rule  \:  \\  \to \sf \: differentiate \: numenator \: and \: denomenator \:  \\  \sf \:  \:  \:  \:  \:  \: both \:  \\  \\  \to \: \red{\lim_{x \to a} }\green{ \frac{ \cos x -  \cos a}{ \cot  x -  \cot a} } \\  \:  \\ \bf after \: differentiation \:  \\  \\  \to \: \blue{\lim_{x \to a} }\pink{ \frac{  - \sin x }{  -  { \csc}^{2}x } } \\  \because \:  \boxed{ \csc \theta =  \frac{1}{ \sin \theta} } \\  \therefore \\  \to \: \purple{\lim_{x \to a} } \:  \: \red{ \frac{  \sin x }{   \frac{1}{ { \sin}^{2}x }  } }\:  \\   \\  \to \: {\lim_{x \to a} } \:  \: \red{  { \sin}^{3} x} \:  \\ \bf \red{ Taking  \: limit \: } \\  \\  \to \:  { \sin}^{3} a \\  \\ \boxed{ \red{\lim_{x \to a} } \bigg \{\green{ \frac{ \cos x -  \cos a}{ \cot  x -  \cot a} }  \bigg \} \:  \:  =  { \sin}^{3} a}

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