Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of sin(2/x)
-
Limit of (-1+x)^2/(-1+x^2)
-
Limit of x-x^2/3
-
Limit of sin(3*x^2)/x^2
- Integral of d{x}:
-
x^3/(2+x^2)
- Graphing y =:
- x^3/(2+x^2)
-
Identical expressions
- x^ three /(two +x^ two)
- x cubed divide by (2 plus x squared )
- x to the power of three divide by (two plus x to the power of two)
- x3/(2+x2)
- x3/2+x2
- x³/(2+x²)
- x to the power of 3/(2+x to the power of 2)
- x^3/2+x^2
- x^3 divide by (2+x^2)
-
Similar expressions
- x^3/(2-x^2)
Limit of the function x^3/(2+x^2)
The solution
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/ 3 \
| x |
lim |------|
x->oo| 2|
\2 + x /
$$\lim_{x \to \infty}\left(\frac{x^{3}}{x^{2} + 2}\right)$$
Limit(x^3/(2 + x^2), x, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{x \to \infty}\left(\frac{x^{3}}{x^{2} + 2}\right)$$
Let's divide numerator and denominator by x^3:
$$\lim_{x \to \infty}\left(\frac{x^{3}}{x^{2} + 2}\right)$$ =
$$\lim_{x \to \infty} \frac{1}{\frac{1}{x} + \frac{2}{x^{3}}}$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty} \frac{1}{\frac{1}{x} + \frac{2}{x^{3}}} = \lim_{u \to 0^+} \frac{1}{2 u^{3} + u}$$
=
$$\frac{1}{2 \cdot 0^{3}} = \infty$$
The final answer:
$$\lim_{x \to \infty}\left(\frac{x^{3}}{x^{2} + 2}\right) = \infty$$
$$\lim_{x \to \infty}\left(\frac{x^{3}}{x^{2} + 2}\right)$$
Let's divide numerator and denominator by x^3:
$$\lim_{x \to \infty}\left(\frac{x^{3}}{x^{2} + 2}\right)$$ =
$$\lim_{x \to \infty} \frac{1}{\frac{1}{x} + \frac{2}{x^{3}}}$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty} \frac{1}{\frac{1}{x} + \frac{2}{x^{3}}} = \lim_{u \to 0^+} \frac{1}{2 u^{3} + u}$$
=
$$\frac{1}{2 \cdot 0^{3}} = \infty$$
The final answer:
$$\lim_{x \to \infty}\left(\frac{x^{3}}{x^{2} + 2}\right) = \infty$$
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
$$\lim_{x \to \infty}\left(x^{2} + 2\right) = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \infty}\left(\frac{x^{3}}{x^{2} + 2}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} x^{3}}{\frac{d}{d x} \left(x^{2} + 2\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{3 x}{2}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{3 x}{2}\right)$$
=
$$\infty$$
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
$$\lim_{x \to \infty} x^{3} = \infty$$
and limit for the denominator is
$$\lim_{x \to \infty}\left(x^{2} + 2\right) = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \infty}\left(\frac{x^{3}}{x^{2} + 2}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} x^{3}}{\frac{d}{d x} \left(x^{2} + 2\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{3 x}{2}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{3 x}{2}\right)$$
=
$$\infty$$
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
$$\lim_{x \to \infty}\left(\frac{x^{3}}{x^{2} + 2}\right) = \infty$$
$$\lim_{x \to 0^-}\left(\frac{x^{3}}{x^{2} + 2}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x^{3}}{x^{2} + 2}\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{x^{3}}{x^{2} + 2}\right) = \frac{1}{3}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x^{3}}{x^{2} + 2}\right) = \frac{1}{3}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{x^{3}}{x^{2} + 2}\right) = -\infty$$
More at x→-oo
$$\lim_{x \to 0^-}\left(\frac{x^{3}}{x^{2} + 2}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x^{3}}{x^{2} + 2}\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{x^{3}}{x^{2} + 2}\right) = \frac{1}{3}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x^{3}}{x^{2} + 2}\right) = \frac{1}{3}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{x^{3}}{x^{2} + 2}\right) = -\infty$$
More at x→-oo
The graph