Mister Exam

Lang:

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

You have entered [src]

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

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

Limit(log(n)/n^2, n, oo, dir='-')

Lopital's rule

We have indeterminateness of type

oo/oo,

i.e. limit for the numerator is

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

and limit for the denominator is

\lim_{n \to \infty} n^{2} = \infty

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

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

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

\lim_{n \to \infty}\left(\frac{1}{2 n^{2}}\right)

\lim_{n \to \infty}\left(\frac{1}{2 n^{2}}\right)

0

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]

0

Expand and simplify

Other limits n→0, -oo, +oo, 1

\lim_{n \to \infty}\left(\frac{\log{\left(n \right)}}{n^{2}}\right) = 0

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

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

\lim_{n \to 1^-}\left(\frac{\log{\left(n \right)}}{n^{2}}\right) = 0

\lim_{n \to 1^+}\left(\frac{\log{\left(n \right)}}{n^{2}}\right) = 0

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

The graph