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)
- Graphing y =:
- x/(log(x)^3+2*x)
-
Identical expressions
- x/(log(x)^ three + two *x)
- x divide by ( logarithm of (x) cubed plus 2 multiply by x)
- x divide by ( logarithm of (x) to the power of three plus two multiply by x)
- x/(log(x)3+2*x)
- x/logx3+2*x
- x/(log(x)³+2*x)
- x/(log(x) to the power of 3+2*x)
- x/(log(x)^3+2x)
- x/(log(x)3+2x)
- x/logx3+2x
- x/logx^3+2x
- x divide by (log(x)^3+2*x)
-
Similar expressions
- x/(log(x)^3-2*x)
Limit of the function x/(log(x)^3+2*x)
The solution
You have entered
[src]
/ x \
lim |-------------|
x->oo| 3 |
\log (x) + 2*x/
$$\lim_{x \to \infty}\left(\frac{x}{2 x + \log{\left(x \right)}^{3}}\right)$$
Limit(x/(log(x)^3 + 2*x), 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
$$\lim_{x \to \infty}\left(2 x + \log{\left(x \right)}^{3}\right) = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \infty}\left(\frac{x}{2 x + \log{\left(x \right)}^{3}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} x}{\frac{d}{d x} \left(2 x + \log{\left(x \right)}^{3}\right)}\right)$$
=
$$\lim_{x \to \infty} \frac{1}{2 + \frac{3 \log{\left(x \right)}^{2}}{x}}$$
=
$$\lim_{x \to \infty} \frac{1}{2 + \frac{3 \log{\left(x \right)}^{2}}{x}}$$
=
$$\frac{1}{2}$$
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
$$\lim_{x \to \infty}\left(2 x + \log{\left(x \right)}^{3}\right) = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \infty}\left(\frac{x}{2 x + \log{\left(x \right)}^{3}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} x}{\frac{d}{d x} \left(2 x + \log{\left(x \right)}^{3}\right)}\right)$$
=
$$\lim_{x \to \infty} \frac{1}{2 + \frac{3 \log{\left(x \right)}^{2}}{x}}$$
=
$$\lim_{x \to \infty} \frac{1}{2 + \frac{3 \log{\left(x \right)}^{2}}{x}}$$
=
$$\frac{1}{2}$$
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
$$\lim_{x \to \infty}\left(\frac{x}{2 x + \log{\left(x \right)}^{3}}\right) = \frac{1}{2}$$
$$\lim_{x \to 0^-}\left(\frac{x}{2 x + \log{\left(x \right)}^{3}}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x}{2 x + \log{\left(x \right)}^{3}}\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{x}{2 x + \log{\left(x \right)}^{3}}\right) = \frac{1}{2}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x}{2 x + \log{\left(x \right)}^{3}}\right) = \frac{1}{2}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{x}{2 x + \log{\left(x \right)}^{3}}\right) = \frac{1}{2}$$
More at x→-oo
$$\lim_{x \to 0^-}\left(\frac{x}{2 x + \log{\left(x \right)}^{3}}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x}{2 x + \log{\left(x \right)}^{3}}\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{x}{2 x + \log{\left(x \right)}^{3}}\right) = \frac{1}{2}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x}{2 x + \log{\left(x \right)}^{3}}\right) = \frac{1}{2}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{x}{2 x + \log{\left(x \right)}^{3}}\right) = \frac{1}{2}$$
More at x→-oo
The graph