Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
- Limit of (1/factorial(x))^(1/x)
- Limit of (1+cos(3*x))^(1/cos(x))
-
Limit of (1-cos(2*x))/(1-cos(4*x))
-
Limit of (1-4*x)/(-2+5*x)
- Graphing y =:
- (4-x^3)/x^2
- Derivative of:
-
(4-x^3)/x^2
-
Identical expressions
- (four -x^ three)/x^ two
- (4 minus x cubed ) divide by x squared
- (four minus x to the power of three) divide by x to the power of two
- (4-x3)/x2
- 4-x3/x2
- (4-x³)/x²
- (4-x to the power of 3)/x to the power of 2
- 4-x^3/x^2
- (4-x^3) divide by x^2
-
Similar expressions
- (4+x^3)/x^2
Limit of the function (4-x^3)/x^2
The solution
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/ 3\
|4 - x |
lim |------|
x->oo| 2 |
\ x /
$$\lim_{x \to \infty}\left(\frac{4 - x^{3}}{x^{2}}\right)$$
Limit((4 - x^3)/x^2, x, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{x \to \infty}\left(\frac{4 - x^{3}}{x^{2}}\right)$$
Let's divide numerator and denominator by x^3:
$$\lim_{x \to \infty}\left(\frac{4 - x^{3}}{x^{2}}\right)$$ =
$$\lim_{x \to \infty}\left(\frac{-1 + \frac{4}{x^{3}}}{\frac{1}{x}}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{-1 + \frac{4}{x^{3}}}{\frac{1}{x}}\right) = \lim_{u \to 0^+}\left(\frac{4 u^{3} - 1}{u}\right)$$
=
$$\frac{-1 + 4 \cdot 0^{3}}{0} = -\infty$$
The final answer:
$$\lim_{x \to \infty}\left(\frac{4 - x^{3}}{x^{2}}\right) = -\infty$$
$$\lim_{x \to \infty}\left(\frac{4 - x^{3}}{x^{2}}\right)$$
Let's divide numerator and denominator by x^3:
$$\lim_{x \to \infty}\left(\frac{4 - x^{3}}{x^{2}}\right)$$ =
$$\lim_{x \to \infty}\left(\frac{-1 + \frac{4}{x^{3}}}{\frac{1}{x}}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{-1 + \frac{4}{x^{3}}}{\frac{1}{x}}\right) = \lim_{u \to 0^+}\left(\frac{4 u^{3} - 1}{u}\right)$$
=
$$\frac{-1 + 4 \cdot 0^{3}}{0} = -\infty$$
The final answer:
$$\lim_{x \to \infty}\left(\frac{4 - x^{3}}{x^{2}}\right) = -\infty$$
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to \infty}\left(4 - x^{3}\right) = -\infty$$
and limit for the denominator is
$$\lim_{x \to \infty} x^{2} = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \infty}\left(\frac{4 - x^{3}}{x^{2}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(4 - x^{3}\right)}{\frac{d}{d x} x^{2}}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{3 x}{2}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{3 x}{2}\right)$$
=
$$-\infty$$
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
$$\lim_{x \to \infty}\left(4 - x^{3}\right) = -\infty$$
and limit for the denominator is
$$\lim_{x \to \infty} x^{2} = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \infty}\left(\frac{4 - x^{3}}{x^{2}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(4 - x^{3}\right)}{\frac{d}{d x} x^{2}}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{3 x}{2}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{3 x}{2}\right)$$
=
$$-\infty$$
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
$$\lim_{x \to \infty}\left(\frac{4 - x^{3}}{x^{2}}\right) = -\infty$$
$$\lim_{x \to 0^-}\left(\frac{4 - x^{3}}{x^{2}}\right) = \infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{4 - x^{3}}{x^{2}}\right) = \infty$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{4 - x^{3}}{x^{2}}\right) = 3$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{4 - x^{3}}{x^{2}}\right) = 3$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{4 - x^{3}}{x^{2}}\right) = \infty$$
More at x→-oo
$$\lim_{x \to 0^-}\left(\frac{4 - x^{3}}{x^{2}}\right) = \infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{4 - x^{3}}{x^{2}}\right) = \infty$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{4 - x^{3}}{x^{2}}\right) = 3$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{4 - x^{3}}{x^{2}}\right) = 3$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{4 - x^{3}}{x^{2}}\right) = \infty$$
More at x→-oo
The graph