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x*e^(4*x)
Limit of the function/ x*e^(4*x)

Limit of the function x*e^(4*x)

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You have entered [src] 
      /   4*x\
 lim  \x*E   /
x->-oo
lim⁡x→−∞(e4xx)\lim_{x \to -\infty}\left(e^{4 x} x\right) 
Limit(x*E^(4*x), x, -oo)
Lopital's rule
We have indeterminateness of type
-oo/oo,

i.e. limit for the numerator is
lim⁡x→−∞x=−∞\lim_{x \to -\infty} x = -\infty
and limit for the denominator is
lim⁡x→−∞e−4x=∞\lim_{x \to -\infty} e^{- 4 x} = \infty
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡x→−∞(e4xx)\lim_{x \to -\infty}\left(e^{4 x} x\right)
=
Let's transform the function under the limit a few
lim⁡x→−∞(xe4x)\lim_{x \to -\infty}\left(x e^{4 x}\right)
=
lim⁡x→−∞(ddxxddxe−4x)\lim_{x \to -\infty}\left(\frac{\frac{d}{d x} x}{\frac{d}{d x} e^{- 4 x}}\right)
=
lim⁡x→−∞(−e4x4)\lim_{x \to -\infty}\left(- \frac{e^{4 x}}{4}\right)
=
lim⁡x→−∞(−e4x4)\lim_{x \to -\infty}\left(- \frac{e^{4 x}}{4}\right)
=
00
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
02468-8-6-4-2-1010-25000000000000000002500000000000000000
Plot the graph
Rapid solution [src] 
0
00
Expand and simplify
Other limits x→0, -oo, +oo, 1
lim⁡x→−∞(e4xx)=0\lim_{x \to -\infty}\left(e^{4 x} x\right) = 0
lim⁡x→∞(e4xx)=∞\lim_{x \to \infty}\left(e^{4 x} x\right) = \infty
More at x→oo
lim⁡x→0−(e4xx)=0\lim_{x \to 0^-}\left(e^{4 x} x\right) = 0
More at x→0 from the left
lim⁡x→0+(e4xx)=0\lim_{x \to 0^+}\left(e^{4 x} x\right) = 0
More at x→0 from the right
lim⁡x→1−(e4xx)=e4\lim_{x \to 1^-}\left(e^{4 x} x\right) = e^{4}
More at x→1 from the left
lim⁡x→1+(e4xx)=e4\lim_{x \to 1^+}\left(e^{4 x} x\right) = e^{4}
More at x→1 from the right
The graph
Limit of the function x*e^(4*x)
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