Limit of the function log(factorial(x))/x

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     /log(x!)\
 lim |-------|
x->oo\   x   /

\lim_{x \to \infty}\left(\frac{\log{\left(x! \right)}}{x}\right)

Limit(log(factorial(x))/x, x, oo, dir='-')

Lopital's rule

We have indeterminateness of type

oo/oo,

i.e. limit for the numerator is

\lim_{x \to \infty} \log{\left(x! \right)} = \infty

and limit for the denominator is

\lim_{x \to \infty} x = \infty

Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.

\lim_{x \to \infty}\left(\frac{\log{\left(x! \right)}}{x}\right)

\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \log{\left(x! \right)}}{\frac{d}{d x} x}\right)

\lim_{x \to \infty}\left(\frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{x!}\right)

\lim_{x \to \infty}\left(\frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{x!}\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)

Rapid solution [src]

oo

\infty

Expand and simplify

Other limits x→0, -oo, +oo, 1

\lim_{x \to \infty}\left(\frac{\log{\left(x! \right)}}{x}\right) = \infty

\lim_{x \to 0^-}\left(\frac{\log{\left(x! \right)}}{x}\right) = - \gamma

\lim_{x \to 0^+}\left(\frac{\log{\left(x! \right)}}{x}\right) = - \gamma

\lim_{x \to 1^-}\left(\frac{\log{\left(x! \right)}}{x}\right) = 0

\lim_{x \to 1^+}\left(\frac{\log{\left(x! \right)}}{x}\right) = 0

\lim_{x \to -\infty}\left(\frac{\log{\left(x! \right)}}{x}\right) = 0