First slide

Introduction to limits

Question

Among (i) $\lim_{x \to \infty} \sec^{- 1} (\frac{x}{sinx})$ and (ii) $\lim_{x \to \infty} \sec^{- 1} (\frac{sinx}{x})$ ,

Moderate

A

(i) exists, (ii) does not exist
B

(i) does not exist, (ii) exists
C

both (i) and (ii) exist
D

neither (i) nor (ii) exists

Solution

$\begin{array}{l} i . \lim_{x \to \infty} \sec^{- 1} (\frac{x}{sinx}) = \sec^{- 1} (\frac{\infty}{\sin \infty}) \\ = \sec^{- 1} (\frac{\infty}{any value between - 1 to 1}) \\ = \sec^{- 1} (\pm \infty) = \frac{π}{2} \end{array}$
$\begin{array}{l} ii . \lim_{x \to \infty} \sec^{- 1} (\frac{\sin x}{x}) = \sec^{- 1} (\frac{\sin \infty}{\infty}) \\ = \sec^{- 1} (\frac{any value between - 1 to 1}{\infty}) \\ = \sec^{- 1} 0 not defined \\ Hence, (i) exists but (ii) does not exist . \end{array}$

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