What is the limit as x approaches 0 of (1+x/5)^{1/x} ?

The limit as x approaches 0 of (1+x/5)^{1/x} is \sqrt[5]{e}

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limit as x approaches 0 of (1+x/5)^{1/x}

x \to 0 lim ((1 + \frac{x}{5})^{\frac{1}{x}})

5 e

Decimal

1.22140 \dots

Solution steps

x \to 0 lim ((1 + \frac{x}{5})^{\frac{1}{x}})

Apply exponent rules

= x \to 0 lim (e^{\frac{1}{x} l n (1 + \frac{x}{5})})

Apply the Limit Chain Rule:

5 e

= 5 e

x \to 6 lim (2 x^{2} + 4)

x \to 0 - lim (x^{4} sin (\frac{1}{x}))

x \to - 2 lim (\frac{x ^{2}}{x + 2})

x \to - 1 - lim (\frac{- 5 - x}{2 x - 2})

x \to 0 - lim (\frac{1}{x} + \frac{1}{x ^{2}})

What is the limit as x approaches 0 of (1+x/5)^{1/x} ?
The limit as x approaches 0 of (1+x/5)^{1/x} is \sqrt[5]{e}