Function Expression :
\[f(x)=\frac{2(\sqrt{x}-1 )}{x-1} \]Domain
\[\left[0, 1\right[ \cup \left]1, \infty\right[ \]Limits
\[\lim_{x \overset{>}{\rightarrow0} }f(x) = 2 \]\[\lim_{x \overset{<}{\rightarrow1} }f(x) = 1 \]
\[\lim_{x \overset{>}{\rightarrow1} }f(x) = 1 \]
\[\lim_{x \rightarrow+\infty}f(x) = 0 \]
\[ \]
Derivate
\[f^{\,\prime}(x)=- \frac{2 \left(\sqrt{x} - 1\right)}{\left(x - 1\right)^{2}} + \frac{1}{\sqrt{x} \left(x - 1\right)} \]\[f^{\,\prime}(x)=\frac{2 \sqrt{x} - x - 1}{\sqrt{x} \left(x^{2} - 2 x + 1\right)} \]
\[ \]
Integral
\[F(x) = 4 \sqrt{x} - 4 \log{\left(\sqrt{x} + 1 \right)} \]Sign Table
Variation Table
Plot
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