What is the limit as x approaches 0 of ((1+5x)/(1+2x))^{1/x} ?

The limit as x approaches 0 of ((1+5x)/(1+2x))^{1/x} is e^3

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

limit as x approaches 0 of ((1+5x)/(1+2x))^{1/x}

x \to 0 lim ((\frac{1 + 5 x}{1 + 2 x})^{\frac{1}{x}})

e^{3}

Decimal

20.08553 \dots

Solution steps

x \to 0 lim ((\frac{1 + 5 x}{1 + 2 x})^{\frac{1}{x}})

Apply exponent rules

= x \to 0 lim (e^{\frac{1}{x} l n (\frac{1 + 5 x}{1 + 2 x})})

Apply the Limit Chain Rule:

e^{3}

= e^{3}

What is the limit as x approaches 0 of ((1+5x)/(1+2x))^{1/x} ?
The limit as x approaches 0 of ((1+5x)/(1+2x))^{1/x} is e^3