Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of (1+3*x)^(5/x)
-
Limit of x^2/(-2+sqrt(4+x^2))
-
Limit of ((5+x^2-6*x)/(5+x^2-5*x))^(2+3*x)
-
Limit of (2+x^2+3*x)/(-4+x^2)
- Integral of d{x}:
- sqrt(x)*exp(-x)
- Derivative of:
- sqrt(x)*exp(-x)
-
Identical expressions
- sqrt(x)*exp(-x)
- square root of (x) multiply by exponent of ( minus x)
- √(x)*exp(-x)
- sqrt(x)exp(-x)
- sqrtxexp-x
-
Similar expressions
- sqrt(x)*exp(x)
Limit of the function sqrt(x)*exp(-x)
The solution
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/ ___ -x\ lim \\/ x *e / x->oo
$$\lim_{x \to \infty}\left(\sqrt{x} e^{- x}\right)$$
Limit(sqrt(x)*exp(-x), x, oo, dir='-')
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to \infty} \sqrt{x} = \infty$$
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.
$$\lim_{x \to \infty}\left(\sqrt{x} e^{- x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \sqrt{x}}{\frac{d}{d x} e^{x}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{e^{- x}}{2 \sqrt{x}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{e^{- x}}{2 \sqrt{x}}\right)$$
=
$$0$$
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
$$\lim_{x \to \infty} \sqrt{x} = \infty$$
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.
$$\lim_{x \to \infty}\left(\sqrt{x} e^{- x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \sqrt{x}}{\frac{d}{d x} e^{x}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{e^{- x}}{2 \sqrt{x}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{e^{- x}}{2 \sqrt{x}}\right)$$
=
$$0$$
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
$$\lim_{x \to \infty}\left(\sqrt{x} e^{- x}\right) = 0$$
$$\lim_{x \to 0^-}\left(\sqrt{x} e^{- x}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\sqrt{x} e^{- x}\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\sqrt{x} e^{- x}\right) = e^{-1}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\sqrt{x} e^{- x}\right) = e^{-1}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\sqrt{x} e^{- x}\right) = \infty i$$
More at x→-oo
$$\lim_{x \to 0^-}\left(\sqrt{x} e^{- x}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\sqrt{x} e^{- x}\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\sqrt{x} e^{- x}\right) = e^{-1}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\sqrt{x} e^{- x}\right) = e^{-1}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\sqrt{x} e^{- x}\right) = \infty i$$
More at x→-oo
The graph