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

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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 Notation

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}

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

z \to 2 lim ((\frac{2 z - z ^{2}}{z ^{2} - 4})^{3})

x \to - 2 lim (4 x^{2} + 3 x - 2)

x \to 3 - lim (\frac{4}{x - 3})

x \to 0 lim (cos (2 x) - 1)

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