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

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

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

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

\infty

Solution steps

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

Divide by highest denominator power:

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

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

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

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

x \to \infty lim (4 + 3 + \frac{2}{x} + \frac{1}{x ^{2}}) = 7

= \frac{\infty}{7}

Apply Infinity Property:

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

= \infty

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