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factorial(x)/x^2
Limit of the function/ factorial(x)/x^2

Limit of the function factorial(x)/x^2

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     /x!\
 lim |--|
x->oo| 2|
     \x /
lim⁡x→∞(x!x2)\lim_{x \to \infty}\left(\frac{x!}{x^{2}}\right) 
Limit(factorial(x)/(x^2), x, oo, dir='-')
Lopital's rule
We have indeterminateness of type
oo/oo,

i.e. limit for the numerator is
lim⁡x→∞x!=∞\lim_{x \to \infty} x! = \infty
and limit for the denominator is
lim⁡x→∞x2=∞\lim_{x \to \infty} x^{2} = \infty
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡x→∞(x!x2)\lim_{x \to \infty}\left(\frac{x!}{x^{2}}\right)
=
Let's transform the function under the limit a few
lim⁡x→∞(x!x2)\lim_{x \to \infty}\left(\frac{x!}{x^{2}}\right)
=
lim⁡x→∞(ddxx!ddxx2)\lim_{x \to \infty}\left(\frac{\frac{d}{d x} x!}{\frac{d}{d x} x^{2}}\right)
=
lim⁡x→∞(Γ(x+1)polygamma⁡(0,x+1)2x)\lim_{x \to \infty}\left(\frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{2 x}\right)
=
lim⁡x→∞(ddxΓ(x+1)polygamma⁡(0,x+1)ddx2x)\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{\frac{d}{d x} 2 x}\right)
=
lim⁡x→∞(Γ(x+1)polygamma⁡2(0,x+1)2+Γ(x+1)polygamma⁡(1,x+1)2)\lim_{x \to \infty}\left(\frac{\Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)}}{2} + \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(1,x + 1 \right)}}{2}\right)
=
lim⁡x→∞(Γ(x+1)polygamma⁡2(0,x+1)2+Γ(x+1)polygamma⁡(1,x+1)2)\lim_{x \to \infty}\left(\frac{\Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)}}{2} + \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(1,x + 1 \right)}}{2}\right)
=
∞\infty
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)
The graph
02468-8-6-4-2-1010-100100
Plot the graph
Rapid solution [src] 
oo
∞\infty
Expand and simplify
Other limits x→0, -oo, +oo, 1
lim⁡x→∞(x!x2)=∞\lim_{x \to \infty}\left(\frac{x!}{x^{2}}\right) = \infty
lim⁡x→0−(x!x2)=∞\lim_{x \to 0^-}\left(\frac{x!}{x^{2}}\right) = \infty
More at x→0 from the left
lim⁡x→0+(x!x2)=∞\lim_{x \to 0^+}\left(\frac{x!}{x^{2}}\right) = \infty
More at x→0 from the right
lim⁡x→1−(x!x2)=1\lim_{x \to 1^-}\left(\frac{x!}{x^{2}}\right) = 1
More at x→1 from the left
lim⁡x→1+(x!x2)=1\lim_{x \to 1^+}\left(\frac{x!}{x^{2}}\right) = 1
More at x→1 from the right
lim⁡x→−∞(x!x2)=0\lim_{x \to -\infty}\left(\frac{x!}{x^{2}}\right) = 0
More at x→-oo
The graph
Limit of the function factorial(x)/x^2
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