lim_x→ 5e^x e^5x 5 = lim_x→ 5e^5(e^x 5 1)x 5 = e^5lim_x→ 5e^x 5 1x 5 [Let x 5 = t , then t→ 0 when x → 5 ] ...

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Rationalization Method to Remove Indeterminate Form

limx→ 5ex - e...

Question

$lim x \to 5 \frac{e^{x} - e^{5}}{x - 5}$

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Solution

$lim x \to 5 \frac{e^{x} - e^{5}}{x - 5}$
$= lim x \to 5 \frac{e^{5} (e^{x - 5} - 1)}{x - 5}$
$= e^{5} lim x \to 5 \frac{e^{x - 5} - 1}{x - 5}$
[Let $x - 5 = t$ , then $t \to 0$ when $x \to 5$ ]
$= e^{5} lim t \to 0 \frac{e^{t} - 1}{t}$
$= e^{5} \cdot 1$
$= e^{5}$ .

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