Function Expression :
\[f(x)=2x-1+\frac{1}{e^x-1} \]Domain
\[\left]-\infty, 0\right[ \cup \left]0, \infty\right[ \]Limits
\[\lim_{x \rightarrow-\infty}f(x) = -\infty \]\[\lim_{x \overset{<}{\rightarrow0} }f(x) = -\infty \]
\[\lim_{x \overset{>}{\rightarrow0} }f(x) = +\infty \]
\[\lim_{x \rightarrow+\infty}f(x) = +\infty \]
\[ \]
Derivate
\[f^{\,\prime}(x)=2 - \frac{e^{x}}{\left(e^{x} - 1\right)^{2}} \]\[f^{\,\prime}(x)=2 - \frac{1}{4 \sinh^{2}{\left(\frac{x}{2} \right)}} \]
\[f^{\,\prime}(x)=\frac{8 \sinh^{2}{\left(\frac{x}{2} \right)} - 1}{4 \sinh^{2}{\left(\frac{x}{2} \right)}} \]
Integral
\[F(x) = x^{2} - 2 x + \log{\left(e^{x} - 1 \right)} \]Sign Table
Variation Table
Plot
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