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

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

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

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

e

Decimal Notation

2.71828 \dots

Solution steps

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

Apply exponent rules

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

Apply the Limit Chain Rule:

e

= e

x \to 0 lim (x ln (3 x))

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

x \to \infty lim (3, x)

x \to 0 lim (ln (1 + 2 x))

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

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