First slide

Introduction to limits

Question

$lim_{x \to \infty} \frac{\cot^{- 1} (\sqrt{x + 1} - \sqrt{x})}{\sec^{- 1} \{{(\begin{matrix} 2 x + 1 \\ x - 1 \end{matrix})}^{x}\}}$ is equal to

Moderate

A

1
B

0
C

$\frac{π}{2}$
D

non-existent

Solution

$lim_{x \to \infty} \frac{\cot^{- 1} (\sqrt{x + 1} - \sqrt{x})}{\sec^{- 1} \{{(\frac{2 x + 1}{x - 1})}^{x}\}}$
$= lim_{x \to \infty} \frac{\cot^{- 1} (\frac{1}{\sqrt{x + 1} + \sqrt{x}})}{\sec^{- 1} \{{(2 + \frac{3}{x - 1})}^{x}\}} = \frac{\cot^{- 1} (0)}{\sec^{- 1} (\infty)} = \frac{π / 2}{π / 2} = 1$

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