First slide

Introduction to limits

Question

The value of $lim_{x \to \infty} {(\frac{π}{2} - \tan^{- 1} x)}^{1 / x^{2}}$ , is

Moderate

A

0
B

1
C

-1
D

$e$

Solution

Let $y = lim_{x \to \infty} {(\frac{π}{2} - \tan^{- 1} x)}^{\frac{1}{x}} [0^{0} form]$
$\begin{array}{l} ∴ \log y = lim_{x \to \infty} \frac{1}{x} \log (\frac{π}{2} - \tan^{- 1} x) \\ \Rightarrow \log y = lim_{x \to \infty} \frac{\log (π / 2 - \tan^{- 1} x)}{x} [\frac{\infty}{\infty} form] \end{array}$
$\Rightarrow \log y = lim_{x \to \infty} \frac{(- \frac{1}{1 + x^{2}})}{\frac{π}{2} - \tan^{- 1} x} [Using L'Hospital's Rule]$
$\Rightarrow \log y = lim_{x \to \infty} \frac{\frac{2 x}{{(1 + x^{2})}^{2}}}{(- \frac{1}{1 + x^{2}})} [Using L'Hospital's Rule]$
$\Rightarrow \log y = lim_{x \to \infty} \frac{- 2 x}{1 + x^{2}} = 0 \Rightarrow y = e^{0} = 1$

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