First slide

Introduction to limits

Question

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

Moderate

A

$a + b$
B

$a - b$
C

$e^{a b}$
D

1

Solution

We have,
$\begin{array}{l} lim_{x \to 0} \frac{e^{a x} - e^{b x}}{x} = lim_{x \to 0} \{\frac{(e^{a x} - 1)}{x} - \frac{(e^{b x} - 1)}{x}\} \\ = a lim_{x \to b} \frac{e^{a x} - 1}{a x} - b lim_{x \to 0} \frac{e^{b x} - 1}{b x} = a \times 1 - b \times 1 = a - b \end{array}$

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