Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of (-3*x^2+2*x+7*x^5)/(-5+x+x^5)
-
Limit of (4-x)/(4+x^2-5*x)
-
Limit of -6-3*x+4*x^2
-
Limit of (3-x)^(x/(2-x))
- Sum of series:
-
2^n/factorial(3*n)
-
Identical expressions
- two ^n/factorial(three *n)
- 2 to the power of n divide by factorial(3 multiply by n)
- two to the power of n divide by factorial(three multiply by n)
- 2n/factorial(3*n)
- 2n/factorial3*n
- 2^n/factorial(3n)
- 2n/factorial(3n)
- 2n/factorial3n
- 2^n/factorial3n
- 2^n divide by factorial(3*n)
Limit of the function 2^n/factorial(3*n)
The solution
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/ n \
| 2 |
lim |------|
n->oo\(3*n)!/
$$\lim_{n \to \infty}\left(\frac{2^{n}}{\left(3 n\right)!}\right)$$
Limit(2^n/factorial(3*n), n, oo, dir='-')
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{n \to \infty} 2^{n} = \infty$$
and limit for the denominator is
$$\lim_{n \to \infty} \left(3 n\right)! = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{n \to \infty}\left(\frac{2^{n}}{\left(3 n\right)!}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} 2^{n}}{\frac{d}{d n} \left(3 n\right)!}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{2^{n} \log{\left(2 \right)}}{3 \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{2^{n} \log{\left(2 \right)}}{3 \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)}}\right)$$
=
$$0$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 1 time(s)
oo/oo,
i.e. limit for the numerator is
$$\lim_{n \to \infty} 2^{n} = \infty$$
and limit for the denominator is
$$\lim_{n \to \infty} \left(3 n\right)! = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{n \to \infty}\left(\frac{2^{n}}{\left(3 n\right)!}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} 2^{n}}{\frac{d}{d n} \left(3 n\right)!}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{2^{n} \log{\left(2 \right)}}{3 \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{2^{n} \log{\left(2 \right)}}{3 \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)}}\right)$$
=
$$0$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 1 time(s)
Other limits n→0, -oo, +oo, 1
$$\lim_{n \to \infty}\left(\frac{2^{n}}{\left(3 n\right)!}\right) = 0$$
$$\lim_{n \to 0^-}\left(\frac{2^{n}}{\left(3 n\right)!}\right) = 1$$
More at n→0 from the left
$$\lim_{n \to 0^+}\left(\frac{2^{n}}{\left(3 n\right)!}\right) = 1$$
More at n→0 from the right
$$\lim_{n \to 1^-}\left(\frac{2^{n}}{\left(3 n\right)!}\right) = \frac{1}{3}$$
More at n→1 from the left
$$\lim_{n \to 1^+}\left(\frac{2^{n}}{\left(3 n\right)!}\right) = \frac{1}{3}$$
More at n→1 from the right
$$\lim_{n \to -\infty}\left(\frac{2^{n}}{\left(3 n\right)!}\right) = 0$$
More at n→-oo
$$\lim_{n \to 0^-}\left(\frac{2^{n}}{\left(3 n\right)!}\right) = 1$$
More at n→0 from the left
$$\lim_{n \to 0^+}\left(\frac{2^{n}}{\left(3 n\right)!}\right) = 1$$
More at n→0 from the right
$$\lim_{n \to 1^-}\left(\frac{2^{n}}{\left(3 n\right)!}\right) = \frac{1}{3}$$
More at n→1 from the left
$$\lim_{n \to 1^+}\left(\frac{2^{n}}{\left(3 n\right)!}\right) = \frac{1}{3}$$
More at n→1 from the right
$$\lim_{n \to -\infty}\left(\frac{2^{n}}{\left(3 n\right)!}\right) = 0$$
More at n→-oo