What is the limit as x approaches-infinity of (5x^4+e^{5x})/(x^4) ?

The limit as x approaches-infinity of (5x^4+e^{5x})/(x^4) is 5

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limit as x approaches-infinity of (5x^4+e^{5x})/(x^4)

x \to - \infty lim (\frac{5 x ^{4} + e ^{5 x}}{x ^{4}})

5

Solution steps

x \to - \infty lim (\frac{5 x ^{4} + e ^{5 x}}{x ^{4}})

Factor out

5 x^{4}

from

5 x^{4} + e^{5 x} : 5 x^{4} (1 + \frac{e ^{5 x}}{5 x ^{4}})

= x \to - \infty lim \frac{5 x ^{4} ( 1 + \frac{e ^{5 x}}{5 x ^{4}} )}{x ^{4}}

Simplify

\frac{5 x ^{4} ( 1 + \frac{e ^{5 x}}{5 x ^{4}} )}{x ^{4}} : \frac{e ^{5 x}}{x ^{4}} + 5

= x \to - \infty lim (\frac{e ^{5 x}}{x ^{4}} + 5)

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{e ^{5 x}}{x ^{4}}) + x \to - \infty lim (5)

x \to - \infty lim (\frac{e ^{5 x}}{x ^{4}}) = 0

x \to - \infty lim (5) = 5

= 0 + 5

Simplify

= 5

What is the limit as x approaches-infinity of (5x^4+e^{5x})/(x^4) ?
The limit as x approaches-infinity of (5x^4+e^{5x})/(x^4) is 5