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Find the limit as x goes to 0 of ((1/x) – (1/(ex – 1)))

Evaluate the limit.

    \[ \lim_{x \to 0} \left( \frac{1}{x} - \frac{1}{e^x - 1} \right). \]


We have

    \begin{align*}   \lim_{x \to 0} \left( \frac{1}{x} - \frac{1}{e^x - 1} \right) &= \lim_{x \to 0} \left( \frac{e^x - 1 - x}{x (e^x-1)} \right) \\[9pt]  &= \lim_{x \to 0} \left( \frac{\frac{1}{2}x^2 + o(x^2)}{x^2 + o(x^2)} \right) \\[9pt]  &= \lim_{x \to 0} \left( \frac{1 + \frac{o(x^2)}{x^2}}{2 + \frac{o(x^2)}{x^2}} \right) \\[9pt]  &= \frac{1}{2}. \end{align*}

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