_ x 0 e^ a x -e^ b x x = - Vidyadip

MCQ

$\lim _{x \rightarrow 0} \frac{e^{a x}-e^{b x}}{x}=$

A
$\alpha+\beta$
B
$\frac{1}{\alpha}+\beta$
C
$\alpha^2+\beta^2$
✓
$\alpha-\beta$

✓

Answer

Correct option: D.

$\alpha-\beta$

(D)
$\lim _{x \rightarrow 0} \frac{ e ^{a x}- e ^{\beta x}}{x}=\lim _{x \rightarrow 0} \frac{ e ^{a x}-1- e ^{\beta x}+1}{x}$
$=\alpha \lim _{x \rightarrow 0} \frac{ e ^{\alpha x}-1}{\alpha x}-\beta \lim _{x \rightarrow 0} \frac{ e ^{\beta x}-1}{\beta x}$
$=\alpha-\beta$

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