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