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