Mister Exam
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How to use it?
- Limit of the function:
-
Limit of 10+x^2+3*x^3+8*x
-
Limit of (-2+sqrt(x))/(-4+x^2)
-
Limit of x+2*x^3+5*x^4-x^2/3
-
Limit of (-x^3+2*x+5*x^4)/(1+x^4-8*x^3)
- Factor polynomial:
- x^3+x^5
- Graphing y =:
- x^3+x^5
-
Identical expressions
- x^ three +x^ five
- x cubed plus x to the power of 5
- x to the power of three plus x to the power of five
- x3+x5
- x³+x⁵
- x to the power of 3+x to the power of 5
-
Similar expressions
- x^3-x^5
- (x^3+x^5-4*x)/(3+x^3-x)
Limit of the function x^3+x^5
The solution
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/ 3 5\ lim \x + x / x->oo
$$\lim_{x \to \infty}\left(x^{5} + x^{3}\right)$$
Limit(x^3 + x^5, x, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{x \to \infty}\left(x^{5} + x^{3}\right)$$
Let's divide numerator and denominator by x^5:
$$\lim_{x \to \infty}\left(x^{5} + x^{3}\right)$$ =
$$\lim_{x \to \infty}\left(\frac{1 + \frac{1}{x^{2}}}{\frac{1}{x^{5}}}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{1 + \frac{1}{x^{2}}}{\frac{1}{x^{5}}}\right) = \lim_{u \to 0^+}\left(\frac{u^{2} + 1}{u^{5}}\right)$$
=
$$\frac{0^{2} + 1}{0} = \infty$$
The final answer:
$$\lim_{x \to \infty}\left(x^{5} + x^{3}\right) = \infty$$
$$\lim_{x \to \infty}\left(x^{5} + x^{3}\right)$$
Let's divide numerator and denominator by x^5:
$$\lim_{x \to \infty}\left(x^{5} + x^{3}\right)$$ =
$$\lim_{x \to \infty}\left(\frac{1 + \frac{1}{x^{2}}}{\frac{1}{x^{5}}}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{1 + \frac{1}{x^{2}}}{\frac{1}{x^{5}}}\right) = \lim_{u \to 0^+}\left(\frac{u^{2} + 1}{u^{5}}\right)$$
=
$$\frac{0^{2} + 1}{0} = \infty$$
The final answer:
$$\lim_{x \to \infty}\left(x^{5} + x^{3}\right) = \infty$$
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
$$\lim_{x \to \infty}\left(x^{5} + x^{3}\right) = \infty$$
$$\lim_{x \to 0^-}\left(x^{5} + x^{3}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(x^{5} + x^{3}\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(x^{5} + x^{3}\right) = 2$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(x^{5} + x^{3}\right) = 2$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(x^{5} + x^{3}\right) = -\infty$$
More at x→-oo
$$\lim_{x \to 0^-}\left(x^{5} + x^{3}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(x^{5} + x^{3}\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(x^{5} + x^{3}\right) = 2$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(x^{5} + x^{3}\right) = 2$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(x^{5} + x^{3}\right) = -\infty$$
More at x→-oo
The graph