Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of x^2*(1/3-cos(8*x)/3)
-
Limit of (-2+x^2-x)/(-2+x+3*x^2)
-
Limit of (-1+sin(x))/cos(x)
-
Limit of (-2*sin(x)+sin(2*x))/(x*log(cos(5*x)))
- Integral of d{x}:
- log(x)^3/x
-
Identical expressions
- log(x)^ three /x
- logarithm of (x) cubed divide by x
- logarithm of (x) to the power of three divide by x
- log(x)3/x
- logx3/x
- log(x)³/x
- log(x) to the power of 3/x
- logx^3/x
- log(x)^3 divide by x
Limit of the function log(x)^3/x
The solution
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[src]
/ 3 \
|log (x)|
lim |-------|
x->oo\ x /
$$\lim_{x \to \infty}\left(\frac{\log{\left(x \right)}^{3}}{x}\right)$$
Limit(log(x)^3/x, x, oo, dir='-')
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to \infty} \log{\left(x \right)}^{3} = \infty$$
and limit for the denominator is
$$\lim_{x \to \infty} x = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \infty}\left(\frac{\log{\left(x \right)}^{3}}{x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \log{\left(x \right)}^{3}}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{3 \log{\left(x \right)}^{2}}{x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \log{\left(x \right)}^{2}}{\frac{d}{d x} \frac{x}{3}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{6 \log{\left(x \right)}}{x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{6 \log{\left(x \right)}}{x}\right)$$
=
$$0$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 2 time(s)
oo/oo,
i.e. limit for the numerator is
$$\lim_{x \to \infty} \log{\left(x \right)}^{3} = \infty$$
and limit for the denominator is
$$\lim_{x \to \infty} x = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \infty}\left(\frac{\log{\left(x \right)}^{3}}{x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \log{\left(x \right)}^{3}}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{3 \log{\left(x \right)}^{2}}{x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \log{\left(x \right)}^{2}}{\frac{d}{d x} \frac{x}{3}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{6 \log{\left(x \right)}}{x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{6 \log{\left(x \right)}}{x}\right)$$
=
$$0$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 2 time(s)
The graph
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to \infty}\left(\frac{\log{\left(x \right)}^{3}}{x}\right) = 0$$
$$\lim_{x \to 0^-}\left(\frac{\log{\left(x \right)}^{3}}{x}\right) = \infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\log{\left(x \right)}^{3}}{x}\right) = -\infty$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{\log{\left(x \right)}^{3}}{x}\right) = 0$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\log{\left(x \right)}^{3}}{x}\right) = 0$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\log{\left(x \right)}^{3}}{x}\right) = 0$$
More at x→-oo
$$\lim_{x \to 0^-}\left(\frac{\log{\left(x \right)}^{3}}{x}\right) = \infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\log{\left(x \right)}^{3}}{x}\right) = -\infty$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{\log{\left(x \right)}^{3}}{x}\right) = 0$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\log{\left(x \right)}^{3}}{x}\right) = 0$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\log{\left(x \right)}^{3}}{x}\right) = 0$$
More at x→-oo
The graph