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Limit of the function log(x)^3/x

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     /   3   \
     |log (x)|
 lim |-------|
x->oo\   x   /

\lim_{x \to \infty}\left(\frac{\log{\left(x \right)}^{3}}{x}\right)

Limit(log(x)^3/x, x, oo, dir='-')

Lopital's rule

We have indeterminateness of type

oo/oo,

i.e. limit for the numerator is

\lim_{x \to \infty} \log{\left(x \right)}^{3} = \infty

and limit for the denominator is

\lim_{x \to \infty} x = \infty

Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.

\lim_{x \to \infty}\left(\frac{\log{\left(x \right)}^{3}}{x}\right)

\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \log{\left(x \right)}^{3}}{\frac{d}{d x} x}\right)

\lim_{x \to \infty}\left(\frac{3 \log{\left(x \right)}^{2}}{x}\right)

\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \log{\left(x \right)}^{2}}{\frac{d}{d x} \frac{x}{3}}\right)

\lim_{x \to \infty}\left(\frac{6 \log{\left(x \right)}}{x}\right)

\lim_{x \to \infty}\left(\frac{6 \log{\left(x \right)}}{x}\right)

0

It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 2 time(s)

The graph

Plot the graph

Rapid solution [src]

0

Expand and simplify

Other limits x→0, -oo, +oo, 1

\lim_{x \to \infty}\left(\frac{\log{\left(x \right)}^{3}}{x}\right) = 0

\lim_{x \to 0^-}\left(\frac{\log{\left(x \right)}^{3}}{x}\right) = \infty

\lim_{x \to 0^+}\left(\frac{\log{\left(x \right)}^{3}}{x}\right) = -\infty

\lim_{x \to 1^-}\left(\frac{\log{\left(x \right)}^{3}}{x}\right) = 0

\lim_{x \to 1^+}\left(\frac{\log{\left(x \right)}^{3}}{x}\right) = 0

\lim_{x \to -\infty}\left(\frac{\log{\left(x \right)}^{3}}{x}\right) = 0

The graph