Lim_ x → 0e^3x e^2x/x = lim_x → 0[(e^3x 1/x ) (e^2x 1/x ) ] = lim_x →[3 (e^3x 1/3x ) 2 (e^2x 1/2x ) ] =3 × 1 2 × 1 = 1 ...

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Standard Limits to Remove Indeterminate Form

lim x → 0e3 x...

Question

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

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Solution

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

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