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e^x/x^2

Limit of the function e^x/x^2

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     / x\
     |E |
 lim |--|
x->oo| 2|
     \x /
limx(exx2)\lim_{x \to \infty}\left(\frac{e^{x}}{x^{2}}\right)
Limit(E^x/x^2, x, oo, dir='-')
Lopital's rule
We have indeterminateness of type
oo/oo,

i.e. limit for the numerator is
limxex=\lim_{x \to \infty} e^{x} = \infty
and limit for the denominator is
limxx2=\lim_{x \to \infty} x^{2} = \infty
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
limx(exx2)\lim_{x \to \infty}\left(\frac{e^{x}}{x^{2}}\right)
=
Let's transform the function under the limit a few
limx(exx2)\lim_{x \to \infty}\left(\frac{e^{x}}{x^{2}}\right)
=
limx(ddxexddxx2)\lim_{x \to \infty}\left(\frac{\frac{d}{d x} e^{x}}{\frac{d}{d x} x^{2}}\right)
=
limx(ex2x)\lim_{x \to \infty}\left(\frac{e^{x}}{2 x}\right)
=
limx(ddxex2ddxx)\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \frac{e^{x}}{2}}{\frac{d}{d x} x}\right)
=
limx(ex2)\lim_{x \to \infty}\left(\frac{e^{x}}{2}\right)
=
limx(ex2)\lim_{x \to \infty}\left(\frac{e^{x}}{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
02468-8-6-4-2-10100250
Other limits x→0, -oo, +oo, 1
limx(exx2)=\lim_{x \to \infty}\left(\frac{e^{x}}{x^{2}}\right) = \infty
limx0(exx2)=\lim_{x \to 0^-}\left(\frac{e^{x}}{x^{2}}\right) = \infty
More at x→0 from the left
limx0+(exx2)=\lim_{x \to 0^+}\left(\frac{e^{x}}{x^{2}}\right) = \infty
More at x→0 from the right
limx1(exx2)=e\lim_{x \to 1^-}\left(\frac{e^{x}}{x^{2}}\right) = e
More at x→1 from the left
limx1+(exx2)=e\lim_{x \to 1^+}\left(\frac{e^{x}}{x^{2}}\right) = e
More at x→1 from the right
limx(exx2)=0\lim_{x \to -\infty}\left(\frac{e^{x}}{x^{2}}\right) = 0
More at x→-oo
Rapid solution [src]
oo
\infty
The graph
Limit of the function e^x/x^2