Mister Exam
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How to use it?
- Limit of the function:
-
Limit of (3*x+14*x^2)/(1+2*x+7*x^2)
- Limit of log(factorial(n))/log(n^n)
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Limit of acot(3*x)
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Limit of 15
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Identical expressions
- log(factorial(n))/log(n^n)
- logarithm of (factorial(n)) divide by logarithm of (n to the power of n)
- log(factorial(n))/log(nn)
- logfactorialn/lognn
- logfactorialn/logn^n
- log(factorial(n)) divide by log(n^n)
Limit of the function log(factorial(n))/log(n^n)
The solution
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/log(n!)\
lim |-------|
n->oo| / n\|
\log\n //
$$\lim_{n \to \infty}\left(\frac{\log{\left(n! \right)}}{\log{\left(n^{n} \right)}}\right)$$
Limit(log(factorial(n))/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
$$\lim_{n \to \infty} \log{\left(n^{n} \right)} = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{n \to \infty}\left(\frac{\log{\left(n! \right)}}{\log{\left(n^{n} \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \log{\left(n! \right)}}{\frac{d}{d n} \log{\left(n^{n} \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{\left(\log{\left(n \right)} + 1\right) n!}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{n!}}{\frac{d}{d n} \left(\log{\left(n \right)} + 1\right)}\right)$$
=
$$\lim_{n \to \infty}\left(n \left(\frac{\Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{n!} + \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{n!} - \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{n!^{2}}\right)\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} n}{\frac{d}{d n} \frac{1}{\frac{\Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{n!} + \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{n!} - \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{n!^{2}}}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)}}{n!^{2}} + \frac{2 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{n!^{2}} + \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}}{n!^{2}} - \frac{2 \Gamma^{3}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)}}{n!^{3}} - \frac{2 \Gamma^{3}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{n!^{3}} + \frac{\Gamma^{4}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)}}{n!^{4}}}{- \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{n!} - \frac{3 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{n!} - \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}{\left(2,n + 1 \right)}}{n!} + \frac{3 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{n!^{2}} + \frac{3 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{n!^{2}} - \frac{2 \Gamma^{3}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{n!^{3}}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)}}{n!^{2}} + \frac{2 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{n!^{2}} + \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}}{n!^{2}} - \frac{2 \Gamma^{3}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)}}{n!^{3}} - \frac{2 \Gamma^{3}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{n!^{3}} + \frac{\Gamma^{4}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)}}{n!^{4}}}{- \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{n!} - \frac{3 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{n!} - \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}{\left(2,n + 1 \right)}}{n!} + \frac{3 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{n!^{2}} + \frac{3 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{n!^{2}} - \frac{2 \Gamma^{3}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{n!^{3}}}\right)$$
=
$$1$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 3 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
$$\lim_{n \to \infty} \log{\left(n^{n} \right)} = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{n \to \infty}\left(\frac{\log{\left(n! \right)}}{\log{\left(n^{n} \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \log{\left(n! \right)}}{\frac{d}{d n} \log{\left(n^{n} \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{\left(\log{\left(n \right)} + 1\right) n!}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{n!}}{\frac{d}{d n} \left(\log{\left(n \right)} + 1\right)}\right)$$
=
$$\lim_{n \to \infty}\left(n \left(\frac{\Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{n!} + \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{n!} - \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{n!^{2}}\right)\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} n}{\frac{d}{d n} \frac{1}{\frac{\Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{n!} + \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{n!} - \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{n!^{2}}}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)}}{n!^{2}} + \frac{2 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{n!^{2}} + \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}}{n!^{2}} - \frac{2 \Gamma^{3}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)}}{n!^{3}} - \frac{2 \Gamma^{3}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{n!^{3}} + \frac{\Gamma^{4}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)}}{n!^{4}}}{- \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{n!} - \frac{3 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{n!} - \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}{\left(2,n + 1 \right)}}{n!} + \frac{3 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{n!^{2}} + \frac{3 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{n!^{2}} - \frac{2 \Gamma^{3}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{n!^{3}}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)}}{n!^{2}} + \frac{2 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{n!^{2}} + \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}}{n!^{2}} - \frac{2 \Gamma^{3}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)}}{n!^{3}} - \frac{2 \Gamma^{3}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{n!^{3}} + \frac{\Gamma^{4}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)}}{n!^{4}}}{- \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{n!} - \frac{3 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{n!} - \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}{\left(2,n + 1 \right)}}{n!} + \frac{3 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{n!^{2}} + \frac{3 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{n!^{2}} - \frac{2 \Gamma^{3}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{n!^{3}}}\right)$$
=
$$1$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 3 time(s)