Mister Exam

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

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

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

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

and limit for the denominator is

\lim_{x \to \infty} \sqrt{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)}^{2}}{\sqrt{x}}\right)

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

\lim_{x \to \infty}\left(\frac{4 \log{\left(x \right)}}{\sqrt{x}}\right)

\lim_{x \to \infty}\left(\frac{4 \log{\left(x \right)}}{\sqrt{x}}\right)

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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

Other limits x→0, -oo, +oo, 1

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

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

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

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

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

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

Rapid solution [src]

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Expand and simplify

The graph