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

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

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

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

1

Solution steps

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

Multiply by the conjugate of

1 - x^{2} : \frac{1 - x ^{4}}{1 + x ^{2}}

= x \to 0 lim ((\frac{1 - x ^{4}}{1 + x ^{2}})^{\frac{1}{x}})

Apply exponent rules

= x \to 0 lim (e^{\frac{1}{x} l n (\frac{1 - x ^{4}}{1 + x ^{2}})})

Apply the Limit Chain Rule:

1

= 1

x \to 0 + lim (x sin (ln (x)))

x \to - 8 + lim (\frac{3 - 4 x}{x + 8})

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

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

x \to - 2 lim (\frac{8}{x ^{2} - 4})

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