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

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

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

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

2

Solution steps

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

Expand

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

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

x \to a lim [f (x) \pm g (x)] = x \to a lim f (x) \pm x \to a lim g (x)

With the exception of indeterminate form

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

x \to \infty lim (\frac{1}{x ^{3}}) = 0

x \to \infty lim (\frac{3}{x ^{2}}) = 0

x \to \infty lim (2) = 2

= 0 + 0 + 2

Simplify

= 2

\int \frac{19}{x ^{3} - 27} d x

\frac{d}{d x} (\frac{- sin ^{2} ( x )}{x})

e^{6 x}

y^{^{''}} - 3 y^{^{'}} + 2 y = 1 + 2 e^{3 t}

t^{2} sin (\frac{1}{2} t)

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