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

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

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

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

\frac{1}{e ^{2}}

Decimal

0.13533 \dots

Solution steps

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

Apply exponent rules

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

Apply the Limit Chain Rule:

\frac{1}{e ^{2}}

= \frac{1}{e ^{2}}

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