Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of x^(4/x)
-
Limit of (-6+x^2-x)/(-4+x^2)
-
Limit of (-6+x^2-x)/(-21+x+2*x^2)
-
Limit of e^(3-x)*(-2+x)
- Integral of d{x}:
-
e^(-x)*x^3
-
Identical expressions
- e^(-x)*x^ three
- e to the power of ( minus x) multiply by x cubed
- e to the power of ( minus x) multiply by x to the power of three
- e(-x)*x3
- e-x*x3
- e^(-x)*x³
- e to the power of (-x)*x to the power of 3
- e^(-x)x^3
- e(-x)x3
- e-xx3
- e^-xx^3
-
Similar expressions
- e^(x)*x^3
Limit of the function e^(-x)*x^3
The solution
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/ -x 3\ lim \E *x / x->oo
$$\lim_{x \to \infty}\left(e^{- x} x^{3}\right)$$
Limit(E^(-x)*x^3, x, oo, dir='-')
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to \infty} x^{3} = \infty$$
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.
$$\lim_{x \to \infty}\left(e^{- x} x^{3}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to \infty}\left(x^{3} e^{- x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} x^{3}}{\frac{d}{d x} e^{x}}\right)$$
=
$$\lim_{x \to \infty}\left(3 x^{2} e^{- x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} 3 x^{2}}{\frac{d}{d x} e^{x}}\right)$$
=
$$\lim_{x \to \infty}\left(6 x e^{- x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} 6 x}{\frac{d}{d x} e^{x}}\right)$$
=
$$\lim_{x \to \infty}\left(6 e^{- x}\right)$$
=
$$\lim_{x \to \infty}\left(6 e^{- x}\right)$$
=
$$0$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 3 time(s)
oo/oo,
i.e. limit for the numerator is
$$\lim_{x \to \infty} x^{3} = \infty$$
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.
$$\lim_{x \to \infty}\left(e^{- x} x^{3}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to \infty}\left(x^{3} e^{- x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} x^{3}}{\frac{d}{d x} e^{x}}\right)$$
=
$$\lim_{x \to \infty}\left(3 x^{2} e^{- x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} 3 x^{2}}{\frac{d}{d x} e^{x}}\right)$$
=
$$\lim_{x \to \infty}\left(6 x e^{- x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} 6 x}{\frac{d}{d x} e^{x}}\right)$$
=
$$\lim_{x \to \infty}\left(6 e^{- x}\right)$$
=
$$\lim_{x \to \infty}\left(6 e^{- x}\right)$$
=
$$0$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 3 time(s)
The graph
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to \infty}\left(e^{- x} x^{3}\right) = 0$$
$$\lim_{x \to 0^-}\left(e^{- x} x^{3}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(e^{- x} x^{3}\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(e^{- x} x^{3}\right) = e^{-1}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(e^{- x} x^{3}\right) = e^{-1}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(e^{- x} x^{3}\right) = -\infty$$
More at x→-oo
$$\lim_{x \to 0^-}\left(e^{- x} x^{3}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(e^{- x} x^{3}\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(e^{- x} x^{3}\right) = e^{-1}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(e^{- x} x^{3}\right) = e^{-1}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(e^{- x} x^{3}\right) = -\infty$$
More at x→-oo
The graph