What is the limit as x approaches infinity of (3x^2)/(x+1)-(x^3)/(x^2+1) ?

The limit as x approaches infinity of (3x^2)/(x+1)-(x^3)/(x^2+1) is infinity

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limit as x approaches infinity of (3x^2)/(x+1)-(x^3)/(x^2+1)

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

\infty

Solution steps

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

Simplify

\frac{3 x ^{2}}{x + 1} - \frac{x ^{3}}{x ^{2} + 1} : \frac{2 x ^{4} - x ^{3} + 3 x ^{2}}{( x + 1 ) ( x ^{2} + 1 )}

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

Apply the following algebraic property

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

\frac{2 x ^{4} - x ^{3} + 3 x ^{2}}{( x + 1 ) ( x ^{2} + 1 )} = \frac{x ^{4} ( 2 - \frac{1}{x} + \frac{3}{x ^{2}} )}{( x ( 1 + \frac{1}{x} ) ) ( x ^{2} ( 1 + \frac{1}{x ^{2}} ) )}

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

Simplify

\frac{x ^{4} ( 2 - \frac{1}{x} + \frac{3}{x ^{2}} )}{( x ( 1 + \frac{1}{x} ) ) ( x ^{2} ( 1 + \frac{1}{x ^{2}} ) )} : \frac{\frac{3}{x} - 1 + 2 x}{( \frac{1}{x} + 1 ) ( \frac{1}{x ^{2}} + 1 )}

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

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

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

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

= \frac{\infty}{1}

Apply Infinity Property:

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

= \infty

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

x \to \infty lim ((- 3)^{n})

x \to 0 + lim (3 e^{x} + cos (2 x))

x \to 0 lim (\frac{x + 12}{x + 6})

x \to 3 + lim ((3 - x) \cdot ln (x - 3))

What is the limit as x approaches infinity of (3x^2)/(x+1)-(x^3)/(x^2+1) ?
The limit as x approaches infinity of (3x^2)/(x+1)-(x^3)/(x^2+1) is infinity