First slide

Introduction to limits

Question

The value of $lim_{x \to 0} (\cos x)^{\cot x},$ is

Moderate

A

$e$
B

$\frac{1}{e}$
C

1
D

$- 1$

Solution

We have,
$\begin{array}{l} lim_{x \to 0} (\cos x)^{\cot x} = lim_{x \to 0} (1 + \cos x - 1)^{\cot x} \\ = lim_{x \to 0} {\{1 - 2 \sin^{2} (\frac{x}{2})\}}^{\cot x} = e^{lim_{x \to 0} - 2 \sin^{2} (x / 2) \cdot \cot x} \\ = e^{lim_{x \to 0} - 2 \frac{\sin^{2} (x / 2)}{2 \sin (x / 2) \cos x / 2} = e^{lim_{x \to 0} - \tan (x / 2) \cdot \cos x} = e^{0} = 1} \end{array}$

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