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

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

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

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

\frac{1}{e}

Decimal

0.60653 \dots

Solution steps

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

Apply exponent rule

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

Apply exponent rules

= x \to 0 lim (e^{\frac{1}{x ^{2}} l n (c o s (x))})

Apply the Limit Chain Rule:

\frac{1}{e}

= \frac{1}{e}

x \to - \infty lim (- 2^{x} - 7)

x \to 0 lim (\frac{3 x ^{2}}{1 + x sin ( x ) - cos ( x )})

x \to \infty lim (x \frac{x}{x + 1})

x \to \infty lim (\frac{e ^{x}}{ln ( x ) + 2})

x \to 0 lim (\frac{- x - ln ( 1 - x )}{8 x ^{2}})

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