Function Expression :
\[f(x)=\frac{\sqrt{x+3}}{x-1}-2 \]Domain
\[\left[-3, 1\right[ \cup \left]1, \infty\right[ \]Limits
\[\lim_{x \overset{>}{\rightarrow-3} }f(x) = -2 \]\[\lim_{x \overset{<}{\rightarrow1} }f(x) = -\infty \]
\[\lim_{x \overset{>}{\rightarrow1} }f(x) = +\infty \]
\[\lim_{x \rightarrow+\infty}f(x) = -2 \]
\[ \]
Derivate
\[f^{\,\prime}(x)=\frac{1}{2 \left(x - 1\right) \sqrt{x + 3}} - \frac{\sqrt{x + 3}}{\left(x - 1\right)^{2}} \]\[f^{\,\prime}(x)=\frac{- x - 7}{2 \left(x - 1\right)^{2} \sqrt{x + 3}} \]
\[ \]
Integral
\[F(x) = - 2 x + \begin{cases} 2 \sqrt{x + 3} - 4 \operatorname{acoth}{\left(\frac{\sqrt{x + 3}}{2} \right)} & \text{for}\: \left|{x + 3}\right| > 4 \\2 \sqrt{x + 3} - 4 \operatorname{atanh}{\left(\frac{\sqrt{x + 3}}{2} \right)} & \text{otherwise} \end{cases} \]Sign Table
Variation Table
Plot
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