Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of (-6+x^2-x)/(9+x^2-6*x)
-
Limit of sqrt(2+x)-sqrt(x)
-
Limit of log(1+3*x^2)/(x^3-5*x^2)
- Limit of ((5+6*x)/(-10+x))^(5*x)
-
Identical expressions
- exp(x)/x^ three
- exponent of (x) divide by x cubed
- exponent of (x) divide by x to the power of three
- exp(x)/x3
- expx/x3
- exp(x)/x³
- exp(x)/x to the power of 3
- expx/x^3
- exp(x) divide by x^3
Limit of the function exp(x)/x^3
The solution
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/ x\
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lim |--|
x->oo| 3|
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$$\lim_{x \to \infty}\left(\frac{e^{x}}{x^{3}}\right)$$
Limit(exp(x)/(x^3), x, oo, dir='-')
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to \infty} e^{x} = \infty$$
and limit for the denominator is
$$\lim_{x \to \infty} x^{3} = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \infty}\left(\frac{e^{x}}{x^{3}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to \infty}\left(\frac{e^{x}}{x^{3}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} e^{x}}{\frac{d}{d x} x^{3}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{e^{x}}{3 x^{2}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} e^{x}}{\frac{d}{d x} 3 x^{2}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{e^{x}}{6 x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} e^{x}}{\frac{d}{d x} 6 x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{e^{x}}{6}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{e^{x}}{6}\right)$$
=
$$\infty$$
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
$$\lim_{x \to \infty} e^{x} = \infty$$
and limit for the denominator is
$$\lim_{x \to \infty} x^{3} = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \infty}\left(\frac{e^{x}}{x^{3}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to \infty}\left(\frac{e^{x}}{x^{3}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} e^{x}}{\frac{d}{d x} x^{3}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{e^{x}}{3 x^{2}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} e^{x}}{\frac{d}{d x} 3 x^{2}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{e^{x}}{6 x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} e^{x}}{\frac{d}{d x} 6 x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{e^{x}}{6}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{e^{x}}{6}\right)$$
=
$$\infty$$
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
$$\lim_{x \to \infty}\left(\frac{e^{x}}{x^{3}}\right) = \infty$$
$$\lim_{x \to 0^-}\left(\frac{e^{x}}{x^{3}}\right) = -\infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{e^{x}}{x^{3}}\right) = \infty$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{e^{x}}{x^{3}}\right) = e$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{e^{x}}{x^{3}}\right) = e$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{e^{x}}{x^{3}}\right) = 0$$
More at x→-oo
$$\lim_{x \to 0^-}\left(\frac{e^{x}}{x^{3}}\right) = -\infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{e^{x}}{x^{3}}\right) = \infty$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{e^{x}}{x^{3}}\right) = e$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{e^{x}}{x^{3}}\right) = e$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{e^{x}}{x^{3}}\right) = 0$$
More at x→-oo
The graph