What is the limit as x approaches 0 of (1-sin(3x))^{1/(2x)} ?

The limit as x approaches 0 of (1-sin(3x))^{1/(2x)} is 1/(e^{3/2)}

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limit as x approaches 0 of (1-sin(3x))^{1/(2x)}

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

\frac{1}{e ^{\frac{3}{2}}}

Decimal Notation

0.22313 \dots

Solution steps

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

x \to a - lim f (x) = x \to a + lim f (x) = L

then

x \to a lim f (x) = L

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

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

= \frac{1}{e ^{\frac{3}{2}}}

x \to 4 + lim (x^{2} + 5 x - 35)

x \to \infty lim (3 (x))

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

x \to \infty lim (\frac{x}{a})

x \to \infty lim (4 x (1 - e^{(\frac{6}{x})}))

What is the limit as x approaches 0 of (1-sin(3x))^{1/(2x)} ?
The limit as x approaches 0 of (1-sin(3x))^{1/(2x)} is 1/(e^{3/2)}