Function Expression :
\[f(x)=ln(\frac{x}{x+9} ) \]Domain
\[\left]-\infty, -9\right[ \cup \left]0, \infty\right[ \]Limits
\[\lim_{x \rightarrow-\infty}f(x) = 0 \]\[\lim_{x \overset{<}{\rightarrow-9} }f(x) = +\infty \]
\[\lim_{x \overset{>}{\rightarrow0} }f(x) = -\infty \]
\[\lim_{x \rightarrow+\infty}f(x) = 0 \]
\[ \]
Derivate
\[f^{\,\prime}(x)=\frac{\left(x + 9\right) \left(- \frac{x}{\left(x + 9\right)^{2}} + \frac{1}{x + 9}\right)}{x} \]\[f^{\,\prime}(x)=\frac{9}{x \left(x + 9\right)} \]
\[ \]
Integral
\[F(x) = x \log{\left(\frac{x}{x + 9} \right)} - 9 \log{\left(x + 9 \right)} \]Sign Table
Variation Table
Plot
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