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Limit of the function (1-log(x))/x^2

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     /1 - log(x)\
 lim |----------|
x->oo|     2    |
     \    x     /

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

Limit((1 - log(x))/x^2, x, oo, dir='-')

Lopital's rule

We have indeterminateness of type

-oo/oo,

i.e. limit for the numerator is

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

and limit for the denominator is

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

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

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

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

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

\lim_{x \to \infty}\left(- \frac{1}{2 x^{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 x→0, -oo, +oo, 1

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

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

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

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

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

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

The graph