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Limit of the function 2xe^(-x^2)

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

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

Limit((2*x)*E^(-x^2), x, -oo)

Lopital's rule

We have indeterminateness of type

-oo/oo,

i.e. limit for the numerator is

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

and limit for the denominator is

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

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

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

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

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

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

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

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

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

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

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

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

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

The graph

Limit of the function 2*x*e^(-x^2)