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

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

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limit as x approaches 0+of (1+4x)^{1/x}

x \to 0 + lim ((1 + 4 x)^{\frac{1}{x}})

e^{4}

Decimal Notation

54.59815 \dots

Solution steps

x \to 0 + lim ((1 + 4 x)^{\frac{1}{x}})

Apply exponent rules

= x \to 0 + lim (e^{\frac{1}{x} l n (1 + 4 x)})

Apply the Limit Chain Rule:

e^{4}

= e^{4}

x \to 0 - lim ((1 + 4 x)^{\frac{3}{x}})

x \to 3 lim (x^{3} + 5 x - 1)

x \to 0 lim ((e^{x} + x)^{\frac{7}{x}})

x \to 3 lim (\frac{x ^{2} - 6 x - 9}{x - 3})

x \to - 8 lim (\frac{1}{x + 8})

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