Mister Exam
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How to use it?
- Limit of the function:
-
Limit of (1-4*x)^(1/x)
-
Limit of (-16+x^2+6*x)/(-2-5*x+3*x^2)
-
Limit of (1+x)^(2/3)-(-1+x)^(2/3)
-
Limit of 1/3+x/3
- Sum of series:
-
n^2/factorial(n)
-
Identical expressions
- n^ two /factorial(n)
- n squared divide by factorial(n)
- n to the power of two divide by factorial(n)
- n2/factorial(n)
- n2/factorialn
- n²/factorial(n)
- n to the power of 2/factorial(n)
- n^2/factorialn
- n^2 divide by factorial(n)
Limit of the function n^2/factorial(n)
The solution
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/ 2\
|n |
lim |--|
n->oo\n!/
$$\lim_{n \to \infty}\left(\frac{n^{2}}{n!}\right)$$
Limit(n^2/factorial(n), n, oo, dir='-')
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{n \to \infty} n^{2} = \infty$$
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.
$$\lim_{n \to \infty}\left(\frac{n^{2}}{n!}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} n^{2}}{\frac{d}{d n} n!}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{2 n}{\Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} 2 n}{\frac{d}{d n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{2}{\Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} + \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{2}{\Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} + \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}\right)$$
=
$$0$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 2 time(s)
oo/oo,
i.e. limit for the numerator is
$$\lim_{n \to \infty} n^{2} = \infty$$
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.
$$\lim_{n \to \infty}\left(\frac{n^{2}}{n!}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} n^{2}}{\frac{d}{d n} n!}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{2 n}{\Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} 2 n}{\frac{d}{d n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{2}{\Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} + \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{2}{\Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} + \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}\right)$$
=
$$0$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 2 time(s)
Other limits n→0, -oo, +oo, 1
$$\lim_{n \to \infty}\left(\frac{n^{2}}{n!}\right) = 0$$
$$\lim_{n \to 0^-}\left(\frac{n^{2}}{n!}\right) = 0$$
More at n→0 from the left
$$\lim_{n \to 0^+}\left(\frac{n^{2}}{n!}\right) = 0$$
More at n→0 from the right
$$\lim_{n \to 1^-}\left(\frac{n^{2}}{n!}\right) = 1$$
More at n→1 from the left
$$\lim_{n \to 1^+}\left(\frac{n^{2}}{n!}\right) = 1$$
More at n→1 from the right
$$\lim_{n \to -\infty}\left(\frac{n^{2}}{n!}\right) = \frac{\infty}{\left(-\infty\right)!}$$
More at n→-oo
$$\lim_{n \to 0^-}\left(\frac{n^{2}}{n!}\right) = 0$$
More at n→0 from the left
$$\lim_{n \to 0^+}\left(\frac{n^{2}}{n!}\right) = 0$$
More at n→0 from the right
$$\lim_{n \to 1^-}\left(\frac{n^{2}}{n!}\right) = 1$$
More at n→1 from the left
$$\lim_{n \to 1^+}\left(\frac{n^{2}}{n!}\right) = 1$$
More at n→1 from the right
$$\lim_{n \to -\infty}\left(\frac{n^{2}}{n!}\right) = \frac{\infty}{\left(-\infty\right)!}$$
More at n→-oo