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  • Evaluate  <img alt="\lim _{a \rightarrow \frac{\pi}{4}}=\frac{\sqrt{2} \cos a-1}{\cot a-1}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Clim%20_%7Ba%20%5Crightarrow%20%5Cfrac%7

Evaluate  \lim _{a \rightarrow \frac{\pi}{4}}=\frac{\sqrt{2} \cos a-1}{\cot a-1}

Option: 1

\frac{3}{2}


Option: 2

\frac{1}{2}


Option: 3

1


Option: 4

0


Answers (1)

best_answer

Given   \lim _{a \rightarrow \frac{\pi}{4}}=\frac{\sqrt{2} \cos a-1}{\cot a-1}

By applying limits it takes the indeterminate form

Apply L-Hospitals rule 

=\lim _{a \rightarrow \frac{\pi}{4}} \frac{-\sqrt{2} \sin a}{-\operatorname{cosec}^2 a}

Apply limits

\begin{aligned} & =\lim _{a \rightarrow \frac{\pi}{4}} \frac{-\sqrt{2}\left(\frac{1}{\sqrt{2}}\right)}{\left(\frac{1}{\sqrt{2}}\right)^2} \\ & =\frac{1}{2} \end{aligned}

Hence , \lim _{a \rightarrow \frac{\pi}{4}}=\frac{\sqrt{2} \cos a-1}{\cot a-1}=\frac{1}{2}

 

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