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