What is the limit as x approaches pi of sin(x/2) ?

The limit as x approaches pi of sin(x/2) is 1

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limit as x approaches pi of sin(x/2)

x \to π lim (sin (\frac{x}{2}))

1

Solution steps

x \to π lim (sin (\frac{x}{2}))

Plug in the value

x = π

= sin (\frac{π}{2})

Use the following trivial identity:

sin (\frac{π}{2}) = 1

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}

= 1

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