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

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

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

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

\frac{1}{e ^{5}}

Decimal

0.00673 \dots

Solution steps

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

Apply exponent rules

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

Simplify

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

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

Apply the common limit:

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

= e^{- 5}

Apply exponent rule:

a^{- b} = \frac{1}{a ^{b}}

= \frac{1}{e ^{5}}

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