Function Expression :
\[f(x)=\frac{e^x+1}{e^x-1}+x \]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
\[f^{\,\prime}(x)=1 + \frac{e^{x}}{e^{x} - 1} - \frac{\left(e^{x} + 1\right) e^{x}}{\left(e^{x} - 1\right)^{2}} \]\[f^{\,\prime}(x)=\frac{e^{2 x} - 4 e^{x} + 1}{e^{2 x} - 2 e^{x} + 1} \]
\[ \]
Integral
\[F(x) = \frac{x^{2}}{2} - x + 2 \log{\left(e^{x} - 1 \right)} \]Sign Table
Variation Table
Plot
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