Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of (5+x-3*x^2)/(4-x+2*x^2)
-
Limit of ((3+4*x)/(-1+4*x))^(-3+2*x)
-
Limit of ((5+2*x)/(-1+2*x))^(3-x)
-
Limit of ((1+2*x)/(2+2*x))^x
- Factor polynomial:
- 8+x^3
-
Identical expressions
- eight +x^ three
- 8 plus x cubed
- eight plus x to the power of three
- 8+x3
- 8+x³
- 8+x to the power of 3
-
Similar expressions
- (-6+x^2-x)/(8+x^3)
- (8+x^3-2*x)/(8-2*x+7*x^3)
- (8+x^3)/(x^2+8*x)
- 8-x^3
- (x^2-2*x)/(-8+x^3)
Limit of the function 8+x^3
The solution
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/ 3\ lim \8 + x / x->oo
$$\lim_{x \to \infty}\left(x^{3} + 8\right)$$
Limit(8 + x^3, x, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{x \to \infty}\left(x^{3} + 8\right)$$
Let's divide numerator and denominator by x^3:
$$\lim_{x \to \infty}\left(x^{3} + 8\right)$$ =
$$\lim_{x \to \infty}\left(\frac{1 + \frac{8}{x^{3}}}{\frac{1}{x^{3}}}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{1 + \frac{8}{x^{3}}}{\frac{1}{x^{3}}}\right) = \lim_{u \to 0^+}\left(\frac{8 u^{3} + 1}{u^{3}}\right)$$
=
$$\frac{8 \cdot 0^{3} + 1}{0} = \infty$$
The final answer:
$$\lim_{x \to \infty}\left(x^{3} + 8\right) = \infty$$
$$\lim_{x \to \infty}\left(x^{3} + 8\right)$$
Let's divide numerator and denominator by x^3:
$$\lim_{x \to \infty}\left(x^{3} + 8\right)$$ =
$$\lim_{x \to \infty}\left(\frac{1 + \frac{8}{x^{3}}}{\frac{1}{x^{3}}}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{1 + \frac{8}{x^{3}}}{\frac{1}{x^{3}}}\right) = \lim_{u \to 0^+}\left(\frac{8 u^{3} + 1}{u^{3}}\right)$$
=
$$\frac{8 \cdot 0^{3} + 1}{0} = \infty$$
The final answer:
$$\lim_{x \to \infty}\left(x^{3} + 8\right) = \infty$$
Lopital's rule
There is no sense to apply Lopital's rule to this function since there is no indeterminateness of 0/0 or oo/oo type
The graph
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to \infty}\left(x^{3} + 8\right) = \infty$$
$$\lim_{x \to 0^-}\left(x^{3} + 8\right) = 8$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(x^{3} + 8\right) = 8$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(x^{3} + 8\right) = 9$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(x^{3} + 8\right) = 9$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(x^{3} + 8\right) = -\infty$$
More at x→-oo
$$\lim_{x \to 0^-}\left(x^{3} + 8\right) = 8$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(x^{3} + 8\right) = 8$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(x^{3} + 8\right) = 9$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(x^{3} + 8\right) = 9$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(x^{3} + 8\right) = -\infty$$
More at x→-oo
The graph