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

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

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

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

e^{2}

Decimal Notation

7.38905 \dots

Solution steps

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

Apply exponent rules

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

Apply the Limit Chain Rule:

e^{2}

= e^{2}

(x + 1) \frac{d y}{d x} + y = x ln^{2} (x)

x \to \infty lim (\frac{( ln ( e ^{3 x} + x ) )}{x})

t \frac{d y}{d t} + 2 y = t^{4}

\frac{\partial}{\partial y} (x ln (y) - y ln (x))

\int \frac{x ^{4} - x ^{3} + 2 x ^{2} - 2 x - 8}{x ^{2} + 4} d x

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