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

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

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

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

1

Solution steps

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

Apply exponent rules

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

Apply the Limit Chain Rule:

1

= 1

\frac{d}{d x} (\frac{1 + ln ( x )}{x ^{2} - ln ( x )})

\int \frac{1}{2} x^{2} ln (x) - \frac{x ^{2}}{4} - \frac{x ^{2}}{2} d x

\int cos (x) sin (x) d x

x \to - 5 lim (\frac{∣ x + 5 ∣}{x + 5})

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