lim-x-0-x-e-x-1-2-e-x-1-x-3-

Question Number 153039 by ZiYangLee last updated on 04/Sep/21

\lim_{x \to 0} \frac{x (e^{x} + 1) - 2 (e^{x} - 1)}{x^{3}} = ?

Answered by puissant last updated on 04/Sep/21

= {l i m}_{x \to 0} \frac{x (1 + x + \frac{x^{2}}{2!} + x^{2} ε (x) + 1) - 2 (1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{6} + x^{3} ε (x) - 1)}{x^{3}}

= {l i m}_{x \to 0} \frac{2 x + x^{2} + \frac{x^{3}}{2} - 2 x - x^{2} - \frac{x^{3}}{3} + x^{3} ε (x)}{x^{3}}

= {l i m}_{x \to 0} \frac{x^{3} (\frac{1}{2} - \frac{1}{3} + ε (x))}{x^{3}}; x \to 0 \Rightarrow ε (x) \to 0 .

= \frac{1}{2} - \frac{1}{3} = \frac{1}{6} .

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