Home » Blog » Find the limit as x goes to 0 of ((1/x) – (1/(ex – 1))) Find the limit as x goes to 0 of ((1/x) – (1/(ex – 1))) by RoRi December 22, 2015 Evaluate the limit. We have Related
![\[ \lim_{x \to 0} \left( \frac{1}{x} - \frac{1}{e^x - 1} \right). \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-283ebf432d8c9bab5eac05508db1ee5b_l3.png "Rendered by QuickLaTeX.com")
![\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*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-e83b64db93b092de2f00aa9da025b240_l3.png "Rendered by QuickLaTeX.com")