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

Limit of the function log(n)/n

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You have entered [src] 
     /log(n)\
 lim |------|
n->oo\  n   /
lim⁡n→∞(log⁡(n)n)\lim_{n \to \infty}\left(\frac{\log{\left(n \right)}}{n}\right) 
Limit(log(n)/n, n, oo, dir='-')
Lopital's rule
We have indeterminateness of type
oo/oo,

i.e. limit for the numerator is
lim⁡n→∞log⁡(n)=∞\lim_{n \to \infty} \log{\left(n \right)} = \infty
and limit for the denominator is
lim⁡n→∞n=∞\lim_{n \to \infty} n = \infty
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡n→∞(log⁡(n)n)\lim_{n \to \infty}\left(\frac{\log{\left(n \right)}}{n}\right)
=
lim⁡n→∞(ddnlog⁡(n)ddnn)\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \log{\left(n \right)}}{\frac{d}{d n} n}\right)
=
lim⁡n→∞1n\lim_{n \to \infty} \frac{1}{n}
=
lim⁡n→∞1n\lim_{n \to \infty} \frac{1}{n}
=
00
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
02468-8-6-4-2-1010-2525
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Rapid solution [src] 
0
00
Expand and simplify
Other limits n→0, -oo, +oo, 1
lim⁡n→∞(log⁡(n)n)=0\lim_{n \to \infty}\left(\frac{\log{\left(n \right)}}{n}\right) = 0
lim⁡n→0−(log⁡(n)n)=∞\lim_{n \to 0^-}\left(\frac{\log{\left(n \right)}}{n}\right) = \infty
More at n→0 from the left
lim⁡n→0+(log⁡(n)n)=−∞\lim_{n \to 0^+}\left(\frac{\log{\left(n \right)}}{n}\right) = -\infty
More at n→0 from the right
lim⁡n→1−(log⁡(n)n)=0\lim_{n \to 1^-}\left(\frac{\log{\left(n \right)}}{n}\right) = 0
More at n→1 from the left
lim⁡n→1+(log⁡(n)n)=0\lim_{n \to 1^+}\left(\frac{\log{\left(n \right)}}{n}\right) = 0
More at n→1 from the right
lim⁡n→−∞(log⁡(n)n)=0\lim_{n \to -\infty}\left(\frac{\log{\left(n \right)}}{n}\right) = 0
More at n→-oo
The graph
Limit of the function log(n)/n
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