0
$\begingroup$

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?

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

$\endgroup$
0
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.