What are the asymptote(s) and hole(s), if any, of $f (x) = \frac{ln x}{x}$ $f(x) = lnx/x$ ?

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Answer 1 · 2017-07-19T20:46:17

The domain of the function is $D = (0, + \infty)$ $D=(0,+oo)$
and it's continious in that domain (there are no holes).

We must calculate $lim_{xrarr0^+}f(x)$ so let's do it:

$lim_{xrarr0^+}f(x)=lim_{xrarr0^+}lnx/x=lim_{xrarr0^+}1/x*lnx=(+oo)(-oo)=$

$-oo$

Now because $lim_{xrarr0^+}f(x)=-oo$

the line $x=0$ is a vertical asymptode of $f(x)$

Now lets calculate $lim_{xrarr+oo}f(x)$

$lim_{xrarr+oo}f(x)=lim_{xrarr+oo}lnx/x$

This limit is $(+oo)/(+oo)$ so we can use L'Hôpital's rule, so :

$lim_{xrarr+oo}lnx/x=lim_{xrarr+oo}(1/x)/1=lim_{xrarr+oo}1/x=0$

so the line $y=0$ is a horizontal asymptode of $f(x)$

What are the asymptote(s) and hole(s), if any, of f(x) = lnx/xf(x)=lnxx f(x) = lnx/x?