Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of 7^(1/(-3+x))
-
Limit of (3-3*x^2+4*x^4+6*x^3)/(2*x^2+7*x^4)
-
Limit of ((5+4*x)/(-1+5*x))^(1+3*x)
-
Limit of (-6-x^2-3*x+4*x^3)/(3-x^2+2*x^3)
- Derivative of:
- x*exp(-x^2/2)
-
Identical expressions
- x*exp(-x^ two / two)
- x multiply by exponent of ( minus x squared divide by 2)
- x multiply by exponent of ( minus x to the power of two divide by two)
- x*exp(-x2/2)
- x*exp-x2/2
- x*exp(-x²/2)
- x*exp(-x to the power of 2/2)
- xexp(-x^2/2)
- xexp(-x2/2)
- xexp-x2/2
- xexp-x^2/2
- x*exp(-x^2 divide by 2)
-
Similar expressions
- x*exp(x^2/2)
Limit of the function x*exp(-x^2/2)
The solution
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lim \x*e /
x->oo
$$\lim_{x \to \infty}\left(x e^{\frac{\left(-1\right) x^{2}}{2}}\right)$$
Limit(x*exp((-x^2)/2), x, oo, dir='-')
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
$$\lim_{x \to \infty} e^{\frac{x^{2}}{2}} = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \infty}\left(x e^{\frac{\left(-1\right) x^{2}}{2}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to \infty}\left(x e^{- \frac{x^{2}}{2}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} x}{\frac{d}{d x} e^{\frac{x^{2}}{2}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{e^{- \frac{x^{2}}{2}}}{x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{e^{- \frac{x^{2}}{2}}}{x}\right)$$
=
$$0$$
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)
oo/oo,
i.e. limit for the numerator is
$$\lim_{x \to \infty} x = \infty$$
and limit for the denominator is
$$\lim_{x \to \infty} e^{\frac{x^{2}}{2}} = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \infty}\left(x e^{\frac{\left(-1\right) x^{2}}{2}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to \infty}\left(x e^{- \frac{x^{2}}{2}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} x}{\frac{d}{d x} e^{\frac{x^{2}}{2}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{e^{- \frac{x^{2}}{2}}}{x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{e^{- \frac{x^{2}}{2}}}{x}\right)$$
=
$$0$$
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
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to \infty}\left(x e^{\frac{\left(-1\right) x^{2}}{2}}\right) = 0$$
$$\lim_{x \to 0^-}\left(x e^{\frac{\left(-1\right) x^{2}}{2}}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(x e^{\frac{\left(-1\right) x^{2}}{2}}\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(x e^{\frac{\left(-1\right) x^{2}}{2}}\right) = e^{- \frac{1}{2}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(x e^{\frac{\left(-1\right) x^{2}}{2}}\right) = e^{- \frac{1}{2}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(x e^{\frac{\left(-1\right) x^{2}}{2}}\right) = 0$$
More at x→-oo
$$\lim_{x \to 0^-}\left(x e^{\frac{\left(-1\right) x^{2}}{2}}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(x e^{\frac{\left(-1\right) x^{2}}{2}}\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(x e^{\frac{\left(-1\right) x^{2}}{2}}\right) = e^{- \frac{1}{2}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(x e^{\frac{\left(-1\right) x^{2}}{2}}\right) = e^{- \frac{1}{2}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(x e^{\frac{\left(-1\right) x^{2}}{2}}\right) = 0$$
More at x→-oo
The graph