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

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1 Answer

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You have to find first limx→∞f(x)x=limx→∞1)lnx=0\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 00. 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 mm for \dfrac{f(x)}{x}\dfrac{f(x)}{x}, we would have seeked next the limit of f(x)-mxf(x)-mx Then there are two main cases:

If \lim_{x\to\infty}f(x)-mx=p\lim_{x\to\infty}f(x)-mx=p, there is an oblique asymptote, with equation y=mx+py=mx+p.

If \lim_{x\to\infty}f(x)-mx=\pm\infty\lim_{x\to\infty}f(x)-mx=\pm\infty, there is a parabolic branch in the direction with slope mm.

For functions which have a Taylor’s expansion, setting t=\frac1xt=\frac1x and considering a Taylor’s expansion of f(\frac1t)f(\frac1t) in a neighbourhood of t=0t=0 may yield directly the equation of the asymptote and the position of the curve w.r.t. its asymptote.