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