Mister Exam

Lang:

Limit of the function x^2/log(x)

You have entered [src]

     /   2  \
     |  x   |
 lim |------|
x->oo\log(x)/

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

Limit(x^2/log(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} x^{2} = \infty

and limit for the denominator is

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

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

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

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

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

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

\infty

It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 1 time(s)

The graph

Plot the graph

Rapid solution [src]

oo

\infty

Expand and simplify

Other limits x→0, -oo, +oo, 1

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

\lim_{x \to 0^-}\left(\frac{x^{2}}{\log{\left(x \right)}}\right) = 0

\lim_{x \to 0^+}\left(\frac{x^{2}}{\log{\left(x \right)}}\right) = 0

\lim_{x \to 1^-}\left(\frac{x^{2}}{\log{\left(x \right)}}\right) = -\infty

\lim_{x \to 1^+}\left(\frac{x^{2}}{\log{\left(x \right)}}\right) = \infty

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

The graph