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

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

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

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

e^{4}

Decimal

54.59815 \dots

Solution steps

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

Apply exponent rules

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

Apply the Limit Chain Rule:

e^{4}

= e^{4}

x \to 4 lim (\frac{x - 4}{x + 1})

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

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

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

x \to \infty lim (\frac{24}{x})

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