What is the limit as x approaches infinity of ((\frac{x+1)/(x^3+2))}{(x/(x^3))} ?

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limit as x approaches infinity of ((\frac{x+1)/(x^3+2))}{(x/(x^3))}

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

1

Solution steps

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

Apply the following algebraic property

: a + b = a (1 + \frac{b}{a})

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

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

Simplify

\frac{\frac{x ( 1 + \frac{1}{x} )}{x ^{3} ( 1 + \frac{2}{x ^{3}} )}}{\frac{x}{x ^{3}}} : \frac{1 + \frac{1}{x}}{1 + \frac{2}{x ^{3}}}

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

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

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

= \frac{1}{1}

Simplify

= 1

What is the limit as x approaches infinity of ((\frac{x+1)/(x^3+2))}{(x/(x^3))} ?
The limit as x approaches infinity of ((\frac{x+1)/(x^3+2))}{(x/(x^3))} is 1