What is the limit as x approaches infinity of ((3x^3+12x^2-8x+1))/((x^2+2)(3x-1)) ?

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limit as x approaches infinity of ((3x^3+12x^2-8x+1))/((x^2+2)(3x-1))

x \to \infty lim (\frac{( 3 x ^{3} + 12 x ^{2} - 8 x + 1 )}{( x ^{2} + 2 ) ( 3 x - 1 )})

1

Solution steps

x \to \infty lim (\frac{3 x ^{3} + 12 x ^{2} - 8 x + 1}{( x ^{2} + 2 ) ( 3 x - 1 )})

Apply the following algebraic property

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

\frac{3 x ^{3} + 12 x ^{2} - 8 x + 1}{( x ^{2} + 2 ) ( 3 x - 1 )} = \frac{x ^{3} ( 3 + \frac{12}{x} - \frac{8}{x ^{2}} + \frac{1}{x ^{3}} )}{( x ^{2} ( 1 + \frac{2}{x ^{2}} ) ) ( x ( 3 - \frac{1}{x} ) )}

= x \to \infty lim (\frac{x ^{3} ( 3 + \frac{12}{x} - \frac{8}{x ^{2}} + \frac{1}{x ^{3}} )}{x ^{2} ( 1 + \frac{2}{x ^{2}} ) x ( 3 - \frac{1}{x} )})

Simplify

\frac{x ^{3} ( 3 + \frac{12}{x} - \frac{8}{x ^{2}} + \frac{1}{x ^{3}} )}{( x ^{2} ( 1 + \frac{2}{x ^{2}} ) ) ( x ( 3 - \frac{1}{x} ) )} : \frac{3 + \frac{12}{x} - \frac{8}{x ^{2}} + \frac{1}{x ^{3}}}{( \frac{2}{x ^{2}} + 1 ) ( - \frac{1}{x} + 3 )}

= x \to \infty lim (\frac{3 + \frac{12}{x} - \frac{8}{x ^{2}} + \frac{1}{x ^{3}}}{( \frac{2}{x ^{2}} + 1 ) ( - \frac{1}{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} ( 3 + \frac{12}{x} - \frac{8}{x ^{2}} + \frac{1}{x ^{3}} )}{lim _{x \to \infty} ( ( \frac{2}{x ^{2}} + 1 ) ( - \frac{1}{x} + 3 ) )}

x \to \infty lim (3 + \frac{12}{x} - \frac{8}{x ^{2}} + \frac{1}{x ^{3}}) = 3

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

= \frac{3}{3}

Simplify

= 1

y^{^{'''}} - 9 y^{^{''}} + 27 y^{^{'}} - 27 y = 0

\frac{d}{d x} (arcsin (8 x))

e^{- \frac{5}{t ^{7}}}

ln (2 x - 1)

\int - x^{7} e^{x^{8} - 7} - 2 x d x

What is the limit as x approaches infinity of ((3x^3+12x^2-8x+1))/((x^2+2)(3x-1)) ?
The limit as x approaches infinity of ((3x^3+12x^2-8x+1))/((x^2+2)(3x-1)) is 1