How to find the oblique asymptote of this non rational fuction?

Question

How do I find the oblique asymptote of non rational functions? I've used a software to check this function, and I know there is a oblique asymptote. How do I do it?

$f(x) = x /\ln(x)$

Answer 1

You have to find first $\displaystyle\lim_{x\to\infty}\frac{f(x)}{x}=\lim_{x\to\infty}\frac{1)}{\ln x}=0$.

So there an asymptotic direction with slope $0$. However there is no horizontal asymptote, because it would imply the function has a (finite) limit at infinity.

If we had found a non-zero limit $m$ for $\dfrac{f(x)}{x}$, we would have seeked next the limit of $f(x)-mx$ Then there are two main cases:

• If $\lim_{x\to\infty}f(x)-mx=p$, there is an oblique asymptote, with equation $y=mx+p$.
• If $\lim_{x\to\infty}f(x)-mx=\pm\infty$, there is a parabolic branch in the direction with slope $m$.

For functions which have a Taylor's expansion, setting $t=\frac1x$ and considering a Taylor's expansion of $f(\frac1t)$ in a neighbourhood of $t=0$ may yield directly the equation of the asymptote and the position of the curve w.r.t. its asymptote.