Mister Exam
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- Limit of the function:
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Limit of (-4-7*x+2*x^2)/(4-13*x+3*x^2)
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Limit of 5+3*n
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Limit of (-12+x^2-4*x)/(48+x^2-14*x)
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Identical expressions
- log(n)/factorial(n)
- logarithm of (n) divide by factorial(n)
- logn/factorialn
- log(n) divide by factorial(n)
Limit of the function log(n)/factorial(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)/factorial(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.
$$\lim_{n \to \infty}\left(\frac{\log{\left(n \right)}}{n!}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \log{\left(n \right)}}{\frac{d}{d n} n!}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{1}{n \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{1}{n \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}\right)$$
=
$$0$$
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
$$\lim_{n \to \infty} \log{\left(n \right)} = \infty$$
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.
$$\lim_{n \to \infty}\left(\frac{\log{\left(n \right)}}{n!}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \log{\left(n \right)}}{\frac{d}{d n} n!}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{1}{n \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{1}{n \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}\right)$$
=
$$0$$
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
$$\lim_{n \to \infty}\left(\frac{\log{\left(n \right)}}{n!}\right) = 0$$
$$\lim_{n \to 0^-}\left(\frac{\log{\left(n \right)}}{n!}\right) = -\infty$$
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
$$\lim_{n \to -\infty}\left(\frac{\log{\left(n \right)}}{n!}\right) = \frac{\infty}{\left(-\infty\right)!}$$
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
$$\lim_{n \to -\infty}\left(\frac{\log{\left(n \right)}}{n!}\right) = \frac{\infty}{\left(-\infty\right)!}$$
More at n→-oo