What is the limit as x approaches infinity of (5^{x+1})/(e^x) ?

The limit as x approaches infinity of (5^{x+1})/(e^x) is infinity

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limit as x approaches infinity of (5^{x+1})/(e^x)

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

\infty

Solution steps

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

Simplify

5^{x + 1} : 5^{x} \cdot 5

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

x \to a lim [c \cdot f (x)] = c \cdot x \to a lim f (x)

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

Apply exponent rule:

\frac{a ^{c}}{b ^{c}} = (\frac{a}{b})^{c}

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

Apply the common limit:

x \to \infty lim (a^{x}) = \infty, a > 1

= 5 \cdot \infty

Apply Infinity Property:

c \cdot \infty = \infty

= \infty

What is the limit as x approaches infinity of (5^{x+1})/(e^x) ?
The limit as x approaches infinity of (5^{x+1})/(e^x) is infinity