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Limit of the function/ e^x/x^2

Limit of the function e^x/x^2

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     / x\
     |E |
 lim |--|
x->oo| 2|
     \x /

\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

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

and limit for the denominator is

\lim_{x \to \infty} x^{2} = \infty

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

\lim_{x \to \infty}\left(\frac{e^{x}}{x^{2}}\right)

=
Let's transform the function under the limit a few

\lim_{x \to \infty}\left(\frac{e^{x}}{x^{2}}\right)

\lim_{x \to \infty}\left(\frac{\frac{d}{d x} e^{x}}{\frac{d}{d x} x^{2}}\right)

\lim_{x \to \infty}\left(\frac{e^{x}}{2 x}\right)

\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \frac{e^{x}}{2}}{\frac{d}{d x} x}\right)

\lim_{x \to \infty}\left(\frac{e^{x}}{2}\right)

\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

Plot the graph

Other limits x→0, -oo, +oo, 1

\lim_{x \to \infty}\left(\frac{e^{x}}{x^{2}}\right) = \infty

\lim_{x \to 0^-}\left(\frac{e^{x}}{x^{2}}\right) = \infty

\lim_{x \to 0^+}\left(\frac{e^{x}}{x^{2}}\right) = \infty

\lim_{x \to 1^-}\left(\frac{e^{x}}{x^{2}}\right) = e

\lim_{x \to 1^+}\left(\frac{e^{x}}{x^{2}}\right) = e

\lim_{x \to -\infty}\left(\frac{e^{x}}{x^{2}}\right) = 0

Rapid solution [src]

oo

\infty

Expand and simplify

The graph