What is the limit as x approaches infinity of (sqrt(4x^6-x))/(x^3+2) ?

The limit as x approaches infinity of (sqrt(4x^6-x))/(x^3+2) is 2

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limit as x approaches infinity of (sqrt(4x^6-x))/(x^3+2)

x \to \infty lim (\frac{4 x ^{6} - x}{x ^{3} + 2})

2

Solution steps

x \to \infty lim (\frac{4 x ^{6} - x}{x ^{3} + 2})

Divide by highest denominator power:

\frac{4 - \frac{1}{x ^{5}}}{1 + \frac{2}{x ^{3}}}

= x \to \infty lim \frac{4 - \frac{1}{x ^{5}}}{1 + \frac{2}{x ^{3}}}

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} ( 4 - \frac{1}{x ^{5}} )}{lim _{x \to \infty} ( 1 + \frac{2}{x ^{3}} )}

x \to \infty lim (4 - \frac{1}{x ^{5}}) = 2

x \to \infty lim (1 + \frac{2}{x ^{3}}) = 1

= \frac{2}{1}

Simplify

= 2

What is the limit as x approaches infinity of (sqrt(4x^6-x))/(x^3+2) ?
The limit as x approaches infinity of (sqrt(4x^6-x))/(x^3+2) is 2