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

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

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

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

3 e

Decimal Notation

8.15484 \dots

Solution steps

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

Apply exponent rules

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

Apply the Limit Chain Rule:

3 e

= 3 e

n \to \infty lim (tan (nπ))

x \to 2 lim (\frac{x + 1}{3 x - 6})

x \to 1^{- 1} lim (2 - x)

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

x \to 2 lim (lo g_{3} (x))

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