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)

Popular >

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}}

x \to 1 + lim (\frac{x}{x})

x \to 9 lim (\frac{lo g _{3} ( x ) ^{2} - 4}{lo g _{3} ( x ) - 2})

x \to - 1 lim (4 - x^{2})

n \to \infty lim (\frac{- 8 + \frac{1}{2 ^{n}}}{\frac{7}{2 ^{n}}})

x \to \infty lim (\frac{3 x ^{8} - 17 x ^{3} + 5}{8 x ^{4} + 12 x ^{3} - x})

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)