Evaluate the following limit: _ x 0 e^bx-e^ax x where 0 < a < b. - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\text{e}^\text{bx}-\text{e}^\text{ax}}{\text{x}}$ where 0 < a < b.

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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\text{e}^\text{bx}-\text{e}^\text{ax}}{\text{x}}$
$=\text{b}\lim\limits_{\text{x}\rightarrow0}\frac{\text{e}^\text{ bx}-1}{\text{bx}}-\text{a}\lim\limits_{\text{x}\rightarrow0}\frac{\text{e}^ \text{ ax}-1}{\text{ax}}$
$=\text{b}-\text{a}$

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