Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of (5+x-3*x^2)/(4-x+2*x^2)
-
Limit of (4-x^2)/(3-x^2)
-
Limit of (3+2*x)/(1-5*x)
-
Limit of (1-2*cos(x))/sin(3*x)
- Sum of series:
-
n^(2*n)/factorial(n)
-
Identical expressions
- n^(two *n)/factorial(n)
- n to the power of (2 multiply by n) divide by factorial(n)
- n to the power of (two multiply by n) divide by factorial(n)
- n(2*n)/factorial(n)
- n2*n/factorialn
- n^(2n)/factorial(n)
- n(2n)/factorial(n)
- n2n/factorialn
- n^2n/factorialn
- n^(2*n) divide by factorial(n)
Limit of the function n^(2*n)/factorial(n)
The solution
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/ 2*n\
|n |
lim |----|
n->oo\ n! /
$$\lim_{n \to \infty}\left(\frac{n^{2 n}}{n!}\right)$$
Limit(n^(2*n)/factorial(n), n, oo, dir='-')
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{n \to \infty} n^{2 n} = \infty$$
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.
$$\lim_{n \to \infty}\left(\frac{n^{2 n}}{n!}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} n^{2 n}}{\frac{d}{d n} n!}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{n^{2 n} \left(2 \log{\left(n \right)} + 2\right)}{\Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{n^{2 n} \left(2 \log{\left(n \right)} + 2\right)}{\Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}\right)$$
=
$$\infty$$
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
$$\lim_{n \to \infty} n^{2 n} = \infty$$
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.
$$\lim_{n \to \infty}\left(\frac{n^{2 n}}{n!}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} n^{2 n}}{\frac{d}{d n} n!}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{n^{2 n} \left(2 \log{\left(n \right)} + 2\right)}{\Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{n^{2 n} \left(2 \log{\left(n \right)} + 2\right)}{\Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}\right)$$
=
$$\infty$$
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)
Other limits n→0, -oo, +oo, 1
$$\lim_{n \to \infty}\left(\frac{n^{2 n}}{n!}\right) = \infty$$
$$\lim_{n \to 0^-}\left(\frac{n^{2 n}}{n!}\right) = 1$$
More at n→0 from the left
$$\lim_{n \to 0^+}\left(\frac{n^{2 n}}{n!}\right) = 1$$
More at n→0 from the right
$$\lim_{n \to 1^-}\left(\frac{n^{2 n}}{n!}\right) = 1$$
More at n→1 from the left
$$\lim_{n \to 1^+}\left(\frac{n^{2 n}}{n!}\right) = 1$$
More at n→1 from the right
$$\lim_{n \to -\infty}\left(\frac{n^{2 n}}{n!}\right) = \frac{\infty}{\left(-\infty\right)!}$$
More at n→-oo
$$\lim_{n \to 0^-}\left(\frac{n^{2 n}}{n!}\right) = 1$$
More at n→0 from the left
$$\lim_{n \to 0^+}\left(\frac{n^{2 n}}{n!}\right) = 1$$
More at n→0 from the right
$$\lim_{n \to 1^-}\left(\frac{n^{2 n}}{n!}\right) = 1$$
More at n→1 from the left
$$\lim_{n \to 1^+}\left(\frac{n^{2 n}}{n!}\right) = 1$$
More at n→1 from the right
$$\lim_{n \to -\infty}\left(\frac{n^{2 n}}{n!}\right) = \frac{\infty}{\left(-\infty\right)!}$$
More at n→-oo