Function Expression :
\[f(x)=1-ln(x^2 )\frac{1}{x} \]Domain
\[\left]-\infty, 0\right[ \cup \left]0, \infty\right[ \]Limits
\[\lim_{x \rightarrow-\infty}f(x) = 1 \]\[\lim_{x \overset{<}{\rightarrow0} }f(x) = -\infty \]
\[\lim_{x \overset{>}{\rightarrow0} }f(x) = +\infty \]
\[\lim_{x \rightarrow+\infty}f(x) = 1 \]
\[ \]
Derivate
\[f^{\,\prime}(x)=\frac{\log{\left(x^{2} \right)}}{x^{2}} - \frac{2}{x^{2}} \]\[f^{\,\prime}(x)=\frac{\log{\left(x^{2} \right)} - 2}{x^{2}} \]
\[ \]
Integral
\[F(x) = x - \begin{cases} 0 & \text{for}\: \frac{1}{\left|{x^{2}}\right|} < 1 \wedge \left|{x^{2}}\right| < 1 \\\frac{\log{\left(x^{2} \right)}^{2}}{4} & \text{for}\: \left|{x^{2}}\right| < 1 \\\frac{\log{\left(\frac{1}{x^{2}} \right)}^{2}}{4} & \text{for}\: \frac{1}{\left|{x^{2}}\right|} < 1 \\\frac{{G_{3, 3}^{3, 0}\left(\begin{matrix} & 1, 1, 1 \\0, 0, 0 & \end{matrix} \middle| {x^{2}} \right)}}{2} + \frac{{G_{3, 3}^{0, 3}\left(\begin{matrix} 1, 1, 1 & \\ & 0, 0, 0 \end{matrix} \middle| {x^{2}} \right)}}{2} & \text{otherwise} \end{cases} \]Sign Table
Variation Table
Plot
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