Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of (3*x+14*x^2)/(1+2*x+7*x^2)
- Limit of log(factorial(n))/log(n^n)
-
Limit of acot(3*x)
-
Limit of 15
- Derivative of:
-
x*e^x
- Graphing y =:
- x*e^x
- Integral of d{x}:
-
x*e^x
-
Identical expressions
- x*e^x
- x multiply by e to the power of x
- x*ex
- xe^x
- xex
-
Similar expressions
- -e^(16*x^2)+x*e^(x^2)
- x*e^(x/2)/(x+e^x)
- x*e^x-x*(1+x)
Limit of the function x*e^x
The solution
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/ x\ lim \x*E / x->-oo
$$\lim_{x \to -\infty}\left(e^{x} x\right)$$
Limit(x*E^x, x, -oo)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to -\infty} x = -\infty$$
and limit for the denominator is
$$\lim_{x \to -\infty} e^{- x} = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to -\infty}\left(e^{x} x\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to -\infty}\left(x e^{x}\right)$$
=
$$\lim_{x \to -\infty}\left(\frac{\frac{d}{d x} x}{\frac{d}{d x} e^{- x}}\right)$$
=
$$\lim_{x \to -\infty}\left(- e^{x}\right)$$
=
$$\lim_{x \to -\infty}\left(- e^{x}\right)$$
=
$$0$$
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)
-oo/oo,
i.e. limit for the numerator is
$$\lim_{x \to -\infty} x = -\infty$$
and limit for the denominator is
$$\lim_{x \to -\infty} e^{- x} = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to -\infty}\left(e^{x} x\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to -\infty}\left(x e^{x}\right)$$
=
$$\lim_{x \to -\infty}\left(\frac{\frac{d}{d x} x}{\frac{d}{d x} e^{- x}}\right)$$
=
$$\lim_{x \to -\infty}\left(- e^{x}\right)$$
=
$$\lim_{x \to -\infty}\left(- e^{x}\right)$$
=
$$0$$
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
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to -\infty}\left(e^{x} x\right) = 0$$
$$\lim_{x \to \infty}\left(e^{x} x\right) = \infty$$
More at x→oo
$$\lim_{x \to 0^-}\left(e^{x} x\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(e^{x} x\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(e^{x} x\right) = e$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(e^{x} x\right) = e$$
More at x→1 from the right
$$\lim_{x \to \infty}\left(e^{x} x\right) = \infty$$
More at x→oo
$$\lim_{x \to 0^-}\left(e^{x} x\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(e^{x} x\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(e^{x} x\right) = e$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(e^{x} x\right) = e$$
More at x→1 from the right
The graph