What is the limit as x approaches infinity of (sqrt(x^3+20x))/(10x-2) ?

The limit as x approaches infinity of (sqrt(x^3+20x))/(10x-2) is infinity

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limit as x approaches infinity of (sqrt(x^3+20x))/(10x-2)

x \to \infty lim (\frac{x ^{3} + 20 x}{10 x - 2})

\infty

Solution steps

x \to \infty lim (\frac{x ^{3} + 20 x}{10 x - 2})

Divide by highest denominator power:

\frac{x ^{\frac{1}{2}} \frac{20}{x ^{2}} + 1}{10 - \frac{2}{x}}

= x \to \infty lim \frac{x ^{\frac{1}{2}} \frac{20}{x ^{2}} + 1}{10 - \frac{2}{x}}

x \to a lim [\frac{f ( x )}{g ( x )}] = \frac{lim _{x \to a} f ( x )}{lim _{x \to a} g ( x )}, x \to a lim g (x) \neq = 0

With the exception of indeterminate form

= \frac{lim _{x \to \infty} ( x ^{\frac{1}{2}} \frac{20}{x ^{2}} + 1 )}{lim _{x \to \infty} ( 10 - \frac{2}{x} )}

x \to \infty lim (x^{\frac{1}{2}} \frac{20}{x ^{2}} + 1) = \infty

x \to \infty lim (10 - \frac{2}{x}) = 10

= \frac{\infty}{10}

Apply Infinity Property:

\frac{\infty}{c} = \infty

= \infty

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What is the limit as x approaches infinity of (sqrt(x^3+20x))/(10x-2) ?
The limit as x approaches infinity of (sqrt(x^3+20x))/(10x-2) is infinity