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x*exp(-5*x)
Limit of the function/ x*exp(-5*x)

Limit of the function x*exp(-5*x)

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     /   -5*x\
 lim \x*e    /
x->oo
$$\lim_{x \to \infty}\left(x e^{- 5 x}\right)$$ 
Limit(x*exp(-5*x), x, oo, dir='-')
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
$$\lim_{x \to \infty} e^{5 x} = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \infty}\left(x e^{- 5 x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} x}{\frac{d}{d x} e^{5 x}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{e^{- 5 x}}{5}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{e^{- 5 x}}{5}\right)$$
=
$$0$$
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
$$0$$
Expand and simplify
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to \infty}\left(x e^{- 5 x}\right) = 0$$
$$\lim_{x \to 0^-}\left(x e^{- 5 x}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(x e^{- 5 x}\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(x e^{- 5 x}\right) = e^{-5}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(x e^{- 5 x}\right) = e^{-5}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(x e^{- 5 x}\right) = -\infty$$
More at x→-oo
The graph
Limit of the function x*exp(-5*x)
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