Mister Exam

Other calculators:

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

Limit of the function e^x*x^2

at →
v

For end points:

The graph:

from to

Piecewise:

The solution

You have entered [src] 
      / x  2\
 lim  \E *x /
x->-oo
lim⁡x→−∞(exx2)\lim_{x \to -\infty}\left(e^{x} x^{2}\right) 
Limit(E^x*x^2, x, -oo)
Lopital's rule
We have indeterminateness of type
oo/oo,

i.e. limit for the numerator is
lim⁡x→−∞x2=∞\lim_{x \to -\infty} x^{2} = \infty
and limit for the denominator is
lim⁡x→−∞e−x=∞\lim_{x \to -\infty} e^{- x} = \infty
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡x→−∞(exx2)\lim_{x \to -\infty}\left(e^{x} x^{2}\right)
=
Let's transform the function under the limit a few
lim⁡x→−∞(x2ex)\lim_{x \to -\infty}\left(x^{2} e^{x}\right)
=
lim⁡x→−∞(ddxx2ddxe−x)\lim_{x \to -\infty}\left(\frac{\frac{d}{d x} x^{2}}{\frac{d}{d x} e^{- x}}\right)
=
lim⁡x→−∞(−2xex)\lim_{x \to -\infty}\left(- 2 x e^{x}\right)
=
lim⁡x→−∞(−2xex)\lim_{x \to -\infty}\left(- 2 x e^{x}\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
02468-8-6-4-2-101002500000
Plot the graph
Rapid solution [src] 
0
00
Expand and simplify
Other limits x→0, -oo, +oo, 1
lim⁡x→−∞(exx2)=0\lim_{x \to -\infty}\left(e^{x} x^{2}\right) = 0
lim⁡x→∞(exx2)=∞\lim_{x \to \infty}\left(e^{x} x^{2}\right) = \infty
More at x→oo
lim⁡x→0−(exx2)=0\lim_{x \to 0^-}\left(e^{x} x^{2}\right) = 0
More at x→0 from the left
lim⁡x→0+(exx2)=0\lim_{x \to 0^+}\left(e^{x} x^{2}\right) = 0
More at x→0 from the right
lim⁡x→1−(exx2)=e\lim_{x \to 1^-}\left(e^{x} x^{2}\right) = e
More at x→1 from the left
lim⁡x→1+(exx2)=e\lim_{x \to 1^+}\left(e^{x} x^{2}\right) = e
More at x→1 from the right
The graph
Limit of the function e^x*x^2
    © Mister Exam – Calculator