First slide

Introduction to limits

Question

The value of $lim_{x \to 0} \frac{e^{x} - \cos 2 x - x}{x^{2}},$ is

Moderate

A

2
B

$\frac{1}{2}$
C

$\frac{3}{2}$
D

$\frac{5}{2}$

Solution

$lim_{x \to 0} \frac{e^{x} - \cos 2 x - x}{x^{2}}$
$\begin{array}{l} = lim_{x \to 0} \frac{(e^{x} - 1 - x) + (1 - \cos 2 x)}{x^{2}} \\ = lim_{x \to 0} \frac{(\frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \dots) + 2 \sin^{2} x}{x^{2}} \\ = lim_{x \to 0} \{(\frac{1}{2!} + \frac{x}{3!} + \dots) + 2 {(\frac{\sin x}{x})}^{2}\} = \frac{1}{2!} + 2 \times 1^{2} = \frac{5}{2} \end{array}$

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