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

The limit as x approaches 0 of (1+2x)^{2/x} is e^4

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

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

e^{4}

Decimal Notation

54.59815 \dots

Solution steps

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

Apply exponent rules

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

Apply the Limit Chain Rule:

e^{4}

= e^{4}

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

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

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

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

h \to 5 - lim (5 - h)

What is the limit as x approaches 0 of (1+2x)^{2/x} ?
The limit as x approaches 0 of (1+2x)^{2/x} is e^4