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

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

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limit as x approaches 0-of (1-e^x)^{x^2}

x \to 0 - lim ((1 - e^{x})^{x^{2}})

1

Solution steps

x \to 0 - lim ((1 - e^{x})^{x^{2}})

Apply exponent rules

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

Apply the Limit Chain Rule:

1

= 1

x \to 0 + lim (x^{2} ln^{2} (x))

x \to 0 + lim (x^{2} ln (2 x^{3}))

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

x \to 5 lim (x^{2} - 4 x + 3)

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

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