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 1

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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}})

1

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:

1

= 1

x \to 0 lim (\frac{5 ^{x} - 2 ^{x}}{x})

x \to \infty lim (19)

h \to 0 lim (\frac{ln ( 3 + h ) - ln ( 3 )}{h})

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

x \to 0 + lim (1^{\frac{l n ( 6 )}{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 1