What is the limit as x approaches infinity of (x^2-9)/(x*(x^2+1)) ?

The limit as x approaches infinity of (x^2-9)/(x*(x^2+1)) is 0

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limit as x approaches infinity of (x^2-9)/(x*(x^2+1))

x \to \infty lim (\frac{x ^{2} - 9}{x \cdot ( x ^{2} + 1 )})

0

Solution steps

x \to \infty lim (\frac{x ^{2} - 9}{x ( x ^{2} + 1 )})

Divide by highest denominator power:

\frac{\frac{x ^{2}}{x ^{2} + 1} - \frac{9}{x ^{2} + 1}}{x}

= x \to \infty lim (\frac{\frac{x ^{2}}{x ^{2} + 1} - \frac{9}{x ^{2} + 1}}{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} ( \frac{x ^{2}}{x ^{2} + 1} - \frac{9}{x ^{2} + 1} )}{lim _{x \to \infty} ( x )}

x \to \infty lim (\frac{x ^{2}}{x ^{2} + 1} - \frac{9}{x ^{2} + 1}) = 1

x \to \infty lim (x) = \infty

= \frac{1}{\infty}

Apply Infinity Property:

\frac{c}{\infty} = 0

= 0

x \to \infty lim (\frac{x ^{2} - 5 x + 6}{x - 2})

x \to \infty lim (ln ((1 - \frac{3}{x})^{x}))

x \to 1 lim (3 x^{2} - 2 x)

x \to 0 lim (\frac{1 - c ( 5 x )}{1 - cos ( 3 x )})

x \to \infty lim (\frac{5000}{20 + 30 e ^{- 0.03 x}})

What is the limit as x approaches infinity of (x^2-9)/(x*(x^2+1)) ?
The limit as x approaches infinity of (x^2-9)/(x*(x^2+1)) is 0