First slide

Introduction to limits

Question

$lim_{x \to 0} \frac{1 + x + x^{2} - e^{x}}{x^{2}}$ is equal to

Moderate

A

1
B

0
C

$1 / 2$
D

none of these

Solution

We have
$\begin{array}{l} lim_{x \to 0} \frac{1 + x + x^{2} - x^{x}}{x^{2}} \\ = \underset{x \to 0}{\lim_{_}} \frac{(1 + x + x^{2}) - (1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \dots)}{x^{2}} \\ = lim_{x \to 0} (1 - \frac{1}{2!}) - \frac{x}{3!} - \frac{x^{2}}{4!} + \dots . \\ = (1 - \frac{1}{2}) = \frac{1}{2} \end{array}$

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