Function Expression :
\[f(x)=\frac{1-2x+ln x}{x-1} \]Domain
\[\left]0, 1\right[ \cup \left]1, \infty\right[ \]Limits
\[\lim_{x \overset{>}{\rightarrow0} }f(x) = +\infty \]\[\lim_{x \overset{<}{\rightarrow1} }f(x) = +\infty \]
\[\lim_{x \overset{>}{\rightarrow1} }f(x) = -\infty \]
\[\lim_{x \rightarrow+\infty}f(x) = -2 \]
\[ \]
Derivate
\[f^{\,\prime}(x)=\frac{-2 + \frac{1}{x}}{x - 1} - \frac{\left(-1\right) 2 x + \log{\left(x \right)} + 1}{\left(x - 1\right)^{2}} \]\[f^{\,\prime}(x)=\frac{- x \log{\left(x \right)} + 2 x - 1}{x \left(x^{2} - 2 x + 1\right)} \]
\[ \]
Integral
\[F(x) = - 2 x - 2 \log{\left(x - 1 \right)} + \log{\left(x - 1 \right)} - \operatorname{Li}_{2}\left(x e^{i \pi} - e^{i \pi}\right) \]Sign Table
Variation Table
Plot
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