What is the limit as x approaches infinity of ((-2)x^3+x^2+1)/((-3)x^4+2x^2+1) ?

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limit as x approaches infinity of ((-2)x^3+x^2+1)/((-3)x^4+2x^2+1)

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

0

Solution steps

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

Divide by highest denominator power:

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

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

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

x \to \infty lim (- \frac{2}{x} + \frac{1}{x ^{2}} + \frac{1}{x ^{4}}) = 0

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

= \frac{0}{- 3}

Apply rule

\frac{0}{a} = 0, a \neq = 0

= 0

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

x \to \infty lim (\frac{x + a + x + b}{x})

x \to \infty lim (2^{x} - x)

x \to - 3 + lim (- \frac{x ^{2}}{3 x + 9})

x \to - 3 + lim (9 - x^{2})

What is the limit as x approaches infinity of ((-2)x^3+x^2+1)/((-3)x^4+2x^2+1) ?
The limit as x approaches infinity of ((-2)x^3+x^2+1)/((-3)x^4+2x^2+1) is 0