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

The limit as x approaches 0 of (1+2x)^{-3/x} is 1/(e^6)

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limit as x approaches 0 of (1+2x)^{-3/x}

x \to 0 lim ((1 + 2 x)^{- \frac{3}{x}})

\frac{1}{e ^{6}}

Decimal

0.00247 \dots

Solution steps

x \to 0 lim ((1 + 2 x)^{- \frac{3}{x}})

Apply exponent rules

= x \to 0 lim (e^{- \frac{3}{x} l n (1 + 2 x)})

Apply the Limit Chain Rule:

\frac{1}{e ^{6}}

= \frac{1}{e ^{6}}

What is the limit as x approaches 0 of (1+2x)^{-3/x} ?
The limit as x approaches 0 of (1+2x)^{-3/x} is 1/(e^6)