lim-x-0-x-e-x-2-2-x-

Question Number 121506 by benjo_mathlover last updated on 08/Nov/20

\lim_{x \to 0} {(x + e^{\frac{x}{2}})}^{\frac{2}{x}} ?

Answered by liberty last updated on 09/Nov/20

L = \lim_{x \to 0} {(x + e^{\frac{x}{2}})}^{\frac{2}{x}}

\ln L = \lim_{x \to 0} \frac{2 \ln (x + e^{\frac{x}{2}})}{x}

\ln L = \lim_{x \to 0} \frac{2 (\frac{1 + \frac{1}{2} e^{\frac{x}{2}}}{x + e^{\frac{x}{2}}})}{1}

\ln L = 2 (\frac{3}{2}) = 3 \Rightarrow L = e^{3} . ▴

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