Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of ((3+x)^2+(3-x)^2)/((3-x)^2-(3+x)^2)
-
Limit of (-1+x)/(-2+sqrt(3+x))
-
Limit of (-x+tan(x))/x^3
-
Limit of (-1+sqrt(x))/(-3+x)
- Sum of series:
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log(n)/n
- Graphing y =:
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log(n)/n
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Identical expressions
- log(n)/n
- logarithm of (n) divide by n
- logn/n
- log(n) divide by n
Limit of the function log(n)/n
The solution
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/log(n)\ lim |------| n->oo\ 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
i.e. limit for the numerator is
$$\lim_{n \to \infty} \log{\left(n \right)} = \infty$$
and limit for the denominator is
$$\lim_{n \to \infty} n = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{n \to \infty}\left(\frac{\log{\left(n \right)}}{n}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \log{\left(n \right)}}{\frac{d}{d n} n}\right)$$
=
$$\lim_{n \to \infty} \frac{1}{n}$$
=
$$\lim_{n \to \infty} \frac{1}{n}$$
=
$$0$$
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)
oo/oo,
i.e. limit for the numerator is
$$\lim_{n \to \infty} \log{\left(n \right)} = \infty$$
and limit for the denominator is
$$\lim_{n \to \infty} n = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{n \to \infty}\left(\frac{\log{\left(n \right)}}{n}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \log{\left(n \right)}}{\frac{d}{d n} n}\right)$$
=
$$\lim_{n \to \infty} \frac{1}{n}$$
=
$$\lim_{n \to \infty} \frac{1}{n}$$
=
$$0$$
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
Other limits n→0, -oo, +oo, 1
$$\lim_{n \to \infty}\left(\frac{\log{\left(n \right)}}{n}\right) = 0$$
$$\lim_{n \to 0^-}\left(\frac{\log{\left(n \right)}}{n}\right) = \infty$$
More at n→0 from the left
$$\lim_{n \to 0^+}\left(\frac{\log{\left(n \right)}}{n}\right) = -\infty$$
More at n→0 from the right
$$\lim_{n \to 1^-}\left(\frac{\log{\left(n \right)}}{n}\right) = 0$$
More at n→1 from the left
$$\lim_{n \to 1^+}\left(\frac{\log{\left(n \right)}}{n}\right) = 0$$
More at n→1 from the right
$$\lim_{n \to -\infty}\left(\frac{\log{\left(n \right)}}{n}\right) = 0$$
More at n→-oo
$$\lim_{n \to 0^-}\left(\frac{\log{\left(n \right)}}{n}\right) = \infty$$
More at n→0 from the left
$$\lim_{n \to 0^+}\left(\frac{\log{\left(n \right)}}{n}\right) = -\infty$$
More at n→0 from the right
$$\lim_{n \to 1^-}\left(\frac{\log{\left(n \right)}}{n}\right) = 0$$
More at n→1 from the left
$$\lim_{n \to 1^+}\left(\frac{\log{\left(n \right)}}{n}\right) = 0$$
More at n→1 from the right
$$\lim_{n \to -\infty}\left(\frac{\log{\left(n \right)}}{n}\right) = 0$$
More at n→-oo
The graph