Find the asymtotes to the function: $f(x)={1+ln|x|\over x(1-ln|x|)}$ Find the asymtotes to the function: $f(x)={1+ln|x|\over x(1-ln|x|)}$. I have difficulties with this one when i try to find the limits, i get indefinites that look like (infinity over infinity)over infinity.. Can anyone help me with this?
The important moments in your function's life include $x = 0$ and $\ln |x| = 1$ (so $x = \pm e$), and $x \rightarrow \pm \infty$. 

As we go to $\pm \infty$ we get $1 \pm \ln |x| \approx \ln |x|$ so we reduce the expression to approximately $\ln |x| / (-x \ln |x|)$ which divides to simply $-1/x$, so these asymptotes are the horizontal asymptote $y = 0$.

At $x = 0$ the logarithms again overpower the nearby constants, creating something which goes like $-1/x$. But at $0$ this function has a vertical asymptote.

At $x = \pm e$ you'll have vertical asymptotes as the numerator does not go to zero while the denominator does.

It may be helpful to look at a graph of $x f(x)$ as that has the essential "weird" behavior spelled out for you: WolframAlpha graph. This function has a vertical asymptote at $y = -1$ for $x \rightarrow \pm \infty$, and also has a value of $y=-1$ at $0$.
The important moments in your function's life include $x = 0$ and $\ln |x| = 1$ (so $x = \pm e$), and $x \rightarrow \pm \infty$. 

As we go to $\pm \infty$ we get $1 \pm \ln |x| \approx \ln |x|$ so we reduce the expression to approximately $\ln |x| / (-x \ln |x|)$ which divides to simply $-1/x$, so these asymptotes are the horizontal asymptote $y = 0$.

At $x = 0$ the logarithms again overpower the nearby constants, creating something which goes like $-1/x$. But at $0$ this function has a vertical asymptote.

At $x = \pm e$ you'll have vertical asymptotes as the numerator does not go to zero while the denominator does.

It may be helpful to look at a graph of $x f(x)$ as that has the essential "weird" behavior spelled out for you: WolframAlpha graph. This function has a vertical asymptote at $y = -1$ for $x \rightarrow \pm \infty$, and also has a value of $y=-1$ at $0$.