First slide

Introduction to limits

Question

$lim_{x \to 0} \frac{e^{x^{2}} - \cos x}{x^{2}}$ is equal to

Moderate

A

$\frac{3}{2}$
B

$\frac{1}{2}$
C

$\frac{2}{3}$
D

none of these

Solution

We have,
$\begin{array}{l} lim_{x \to 0} \frac{e^{x^{2}} - \cos x}{x^{2}} = lim_{x \to 0} \{\frac{(e^{x^{2}} - 1)}{x^{2}} + \frac{(1 - \cos x)}{x^{2}}\} \\ = lim_{x \to 0} \frac{e^{x^{2}} - 1}{x^{2}} + lim_{x \to 0} \frac{1 - \cos x}{x^{2}} = 1 + \frac{1}{2} = \frac{3}{2} \end{array}$

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