Function Expression :
\[f(x)=\frac{x-(x+1 )^2. e^{-x}}{x} \]Domain
\[\left]-\infty, 0\right[ \cup \left]0, \infty\right[ \]Limits
\[\lim_{x \rightarrow-\infty}f(x) = +\infty \]\[\lim_{x \overset{<}{\rightarrow0} }f(x) = +\infty \]
\[\lim_{x \overset{>}{\rightarrow0} }f(x) = -\infty \]
\[\lim_{x \rightarrow+\infty}f(x) = 1 \]
\[ \]
Derivate
\[f^{\,\prime}(x)=\frac{\left(- 2 x - 2\right) e^{- x} + \left(x + 1\right)^{2} e^{- x} + 1}{x} - \frac{x - \left(x + 1\right)^{2} e^{- x}}{x^{2}} \]\[f^{\,\prime}(x)=\frac{\left(x^{3} + x^{2} + x + 1\right) e^{- x}}{x^{2}} \]
\[ \]
Integral
\[F(x) = x + x e^{- x} - \operatorname{Ei}{\left(- x \right)} + 3 e^{- x} \]Sign Table
Variation Table
Plot
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