Function Expression :
\[f(x)=x+\frac{e^x}{2(e^x-2 )} \]Domain
\[\left]-\infty, \log{\left]2 \right[}\right[ \cup \left]\log{\left]2 \right[}, \infty\right[ \]Limits
\[\lim_{x \rightarrow-\infty}f(x) = -\infty \]\[\lim_{x \overset{<}{\rightarrow\log{\left(2 \right)}} }f(x) = -\infty \]
\[\lim_{x \overset{>}{\rightarrow\log{\left(2 \right)}} }f(x) = +\infty \]
\[\lim_{x \rightarrow+\infty}f(x) = +\infty \]
\[ \]
Derivate
\[f^{\,\prime}(x)=\frac{1}{2 \left(e^{x} - 2\right)} e^{x} + 1 - \frac{e^{2 x}}{2 \left(e^{x} - 2\right)^{2}} \]\[f^{\,\prime}(x)=\frac{e^{2 x} - 5 e^{x} + 4}{e^{2 x} - 4 e^{x} + 4} \]
\[ \]
Integral
\[F(x) = \frac{x^{2}}{2} + \frac{\log{\left(e^{x} - 2 \right)}}{2} \]Sign Table
Variation Table
Plot
Elapsed Time: 0.0041 seconds