What is the limit as x approaches infinity of (x^2+2x+5)/(e^{3x)} ?

The limit as x approaches infinity of (x^2+2x+5)/(e^{3x)} is 0

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limit as x approaches infinity of (x^2+2x+5)/(e^{3x)}

x \to \infty lim (\frac{x ^{2} + 2 x + 5}{e ^{3 x}})

0

Solution steps

x \to \infty lim (\frac{x ^{2} + 2 x + 5}{e ^{3 x}})

Expand

\frac{x ^{2} + 2 x + 5}{e ^{3 x}} : \frac{x ^{2}}{e ^{3 x}} + \frac{2 x}{e ^{3 x}} + \frac{5}{e ^{3 x}}

= x \to \infty lim (\frac{x ^{2}}{e ^{3 x}} + \frac{2 x}{e ^{3 x}} + \frac{5}{e ^{3 x}})

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{x ^{2}}{e ^{3 x}}) + x \to \infty lim (\frac{2 x}{e ^{3 x}}) + x \to \infty lim (\frac{5}{e ^{3 x}})

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

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

x \to \infty lim (\frac{5}{e ^{3 x}}) = 0

= 0 + 0 + 0

Simplify

= 0

\frac{d}{d x} (e^{x} cos (x) + e^{x} sin (x))

\int \frac{sin ^{2} ( 3 x )}{cos ( 3 x )} d x

\int \frac{1}{( 169 - x ^{2} ) ^{\frac{3}{2}}} d x

1 - x^{3}

6 x^{2}, x^{2} + 5

What is the limit as x approaches infinity of (x^2+2x+5)/(e^{3x)} ?
The limit as x approaches infinity of (x^2+2x+5)/(e^{3x)} is 0