What is the limit as x approaches pi/6 of sin^x(x) ?

The limit as x approaches pi/6 of sin^x(x) is (1/2)^{pi/6}

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limit as x approaches pi/6 of sin^x(x)

x \to \frac{π}{6} lim (sin^{x} (x))

(\frac{1}{2})^{\frac{π}{6}}

Decimal

0.69563 \dots

Solution steps

x \to \frac{π}{6} lim (sin^{x} (x))

Plug in the value

x = \frac{π}{6}

= sin^{\frac{π}{6}} (\frac{π}{6})

Use the following trivial identity:

sin (\frac{π}{6}) = \frac{1}{2}

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= (\frac{1}{2})^{\frac{π}{6}}

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

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

x \to 0 lim (x + e^{2 x})

x \to 2 lim (\frac{x ^{2} - 4}{x + 1})

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

What is the limit as x approaches pi/6 of sin^x(x) ?
The limit as x approaches pi/6 of sin^x(x) is (1/2)^{pi/6}