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

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

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limit as x approaches 0 of (e^x-x)^{3/x}

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

1

Solution steps

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

Apply exponent rules

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

Apply the Limit Chain Rule:

1

= 1

x \to - \frac{π}{2} lim (- 7 x - \frac{π}{2})

x \to 1 lim (\frac{x - 1}{x ^{2} - 8 x + 7})

x \to \infty lim (x \cdot (e^{0 x} - 1))

2 \to 10 lim (2^{2} + 4)

x \to 2 lim (∣ x - 1 ∣)

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