Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of (-3+sqrt(1+4*x))/(-2+x)
-
Limit of x^2+(sqrt(1+3*x)-sqrt(1-2*x))/x
-
Limit of ((11+7*x+7*x^2)/(14+9*x+13*x^2))^(4*x)
-
Limit of (-1+x+6*x^2)/(1/2+x)
- Derivative of:
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x^3+5*x
- Factor polynomial:
- x^3+5*x
-
Identical expressions
- x^ three + five *x
- x cubed plus 5 multiply by x
- x to the power of three plus five multiply by x
- x3+5*x
- x³+5*x
- x to the power of 3+5*x
- x^3+5x
- x3+5x
-
Similar expressions
- (x^3+5*x)/(-10+x)
- x^3-5*x
- 3/2+3*x+4*x^3+5*x^2
- 4*x^3+5*x+6*x^2
- -4*x+2*x^3+5*x^2+x^4/3
Limit of the function x^3+5*x
The solution
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/ 3 \ lim \x + 5*x/ x->oo
$$\lim_{x \to \infty}\left(x^{3} + 5 x\right)$$
Limit(x^3 + 5*x, x, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{x \to \infty}\left(x^{3} + 5 x\right)$$
Let's divide numerator and denominator by x^3:
$$\lim_{x \to \infty}\left(x^{3} + 5 x\right)$$ =
$$\lim_{x \to \infty}\left(\frac{1 + \frac{5}{x^{2}}}{\frac{1}{x^{3}}}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{1 + \frac{5}{x^{2}}}{\frac{1}{x^{3}}}\right) = \lim_{u \to 0^+}\left(\frac{5 u^{2} + 1}{u^{3}}\right)$$
=
$$\frac{5 \cdot 0^{2} + 1}{0} = \infty$$
The final answer:
$$\lim_{x \to \infty}\left(x^{3} + 5 x\right) = \infty$$
$$\lim_{x \to \infty}\left(x^{3} + 5 x\right)$$
Let's divide numerator and denominator by x^3:
$$\lim_{x \to \infty}\left(x^{3} + 5 x\right)$$ =
$$\lim_{x \to \infty}\left(\frac{1 + \frac{5}{x^{2}}}{\frac{1}{x^{3}}}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{1 + \frac{5}{x^{2}}}{\frac{1}{x^{3}}}\right) = \lim_{u \to 0^+}\left(\frac{5 u^{2} + 1}{u^{3}}\right)$$
=
$$\frac{5 \cdot 0^{2} + 1}{0} = \infty$$
The final answer:
$$\lim_{x \to \infty}\left(x^{3} + 5 x\right) = \infty$$
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
$$\lim_{x \to \infty}\left(x^{3} + 5 x\right) = \infty$$
$$\lim_{x \to 0^-}\left(x^{3} + 5 x\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(x^{3} + 5 x\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(x^{3} + 5 x\right) = 6$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(x^{3} + 5 x\right) = 6$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(x^{3} + 5 x\right) = -\infty$$
More at x→-oo
$$\lim_{x \to 0^-}\left(x^{3} + 5 x\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(x^{3} + 5 x\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(x^{3} + 5 x\right) = 6$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(x^{3} + 5 x\right) = 6$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(x^{3} + 5 x\right) = -\infty$$
More at x→-oo
The graph