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

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

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

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

\frac{1}{e}

Decimal

0.36787 \dots

Solution steps

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

Apply exponent rules

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

Apply the Limit Chain Rule:

\frac{1}{e}

= \frac{1}{e}

x \to - \infty lim (\frac{x}{4})

y \to 0 lim (y + y^{3} sin (\frac{x}{y}))

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

x \to π lim (8 e^{\frac{c o s ( x )}{x - π}})

x \to 0 lim (\frac{e ^{x}}{x ^{5}})

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