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Limit of the function xexp(2x)

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      /   2*x\
 lim  \x*e   /
x->-oo

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

Limit(x*exp(2*x), x, -oo)

Lopital's rule

We have indeterminateness of type

-oo/oo,

i.e. limit for the numerator is

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

and limit for the denominator is

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

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

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

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

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

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

0

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)

The graph

Plot the graph

Rapid solution [src]

0

Expand and simplify

Other limits x→0, -oo, +oo, 1

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

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

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

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

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

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

The graph

Limit of the function x*exp(2*x)