Mister Exam

Limit of the function log(n)/sqrt(n)

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The solution

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     /log(n)\
lim |------|
n->oo|  ___ |
\\/ n  /
$$\lim_{n \to \infty}\left(\frac{\log{\left(n \right)}}{\sqrt{n}}\right)$$
Limit(log(n)/sqrt(n), 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
$$\lim_{n \to \infty} \sqrt{n} = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{n \to \infty}\left(\frac{\log{\left(n \right)}}{\sqrt{n}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \log{\left(n \right)}}{\frac{d}{d n} \sqrt{n}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{2}{\sqrt{n}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{2}{\sqrt{n}}\right)$$
=
$$0$$
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
Rapid solution [src]
0
$$0$$
Other limits n→0, -oo, +oo, 1
$$\lim_{n \to \infty}\left(\frac{\log{\left(n \right)}}{\sqrt{n}}\right) = 0$$
$$\lim_{n \to 0^-}\left(\frac{\log{\left(n \right)}}{\sqrt{n}}\right) = \infty i$$
More at n→0 from the left
$$\lim_{n \to 0^+}\left(\frac{\log{\left(n \right)}}{\sqrt{n}}\right) = -\infty$$
More at n→0 from the right
$$\lim_{n \to 1^-}\left(\frac{\log{\left(n \right)}}{\sqrt{n}}\right) = 0$$
More at n→1 from the left
$$\lim_{n \to 1^+}\left(\frac{\log{\left(n \right)}}{\sqrt{n}}\right) = 0$$
More at n→1 from the right
$$\lim_{n \to -\infty}\left(\frac{\log{\left(n \right)}}{\sqrt{n}}\right) = 0$$
More at n→-oo
The graph
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