Limx #8594;0 #160;e3x e2xx=limx #8594;0 #160;e3x 1x e2x 1x=limx #8594;0 #160;3e3x 13x 2e2x 12x=3 #215;1 2 #215;1=1 ...

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Factorization

lim x → 0 e3x...

Question

$\lim_{x \to 0} \frac{e^{3 x} - e^{2 x}}{x}$

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Solution

$\lim_{x \to 0} [\frac{e^{3 x} - e^{2 x}}{x}] = \lim_{x \to 0} [(\frac{e^{3 x} - 1}{x}) - (\frac{e^{2 x} - 1}{x})] = \lim_{x \to 0} [3 (\frac{e^{3 x} - 1}{3 x}) - 2 (\frac{e^{2 x} - 1}{2 x})] = 3 \times 1 - 2 \times 1 = 1$

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