Proving by definition that lim $1/(e^x - 1) = 0$ as $x \rightarrow \infty$

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Proving by definition that lim $1/(e^x - 1) = 0$ as $x \rightarrow \infty$.

My answer is: I have taken $M \geq ln (1+ (1/ \epsilon))$ am I correct?

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If you mean that the function is less than $\epsilon$ for $x > M$, then yes.