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Limit of the function/ x^3+5*x

Limit of the function x^3+5*x

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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

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

Plot the graph

Rapid solution [src]

oo

\infty

Expand and simplify

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

\lim_{x \to 0^+}\left(x^{3} + 5 x\right) = 0

\lim_{x \to 1^-}\left(x^{3} + 5 x\right) = 6

\lim_{x \to 1^+}\left(x^{3} + 5 x\right) = 6

\lim_{x \to -\infty}\left(x^{3} + 5 x\right) = -\infty

The graph