lim-x-pi-sin-x-2-1-x-pi-

Question Number 179366 by mathlove last updated on 28/Oct/22

\lim_{x \to π} \frac{s i n \frac{x}{2} - 1}{x - π} = ?

Commented by mahdipoor last updated on 28/Oct/22

= D_{x} {(s i n \frac{x}{2})}_{x = π} = \frac{1}{2} c o s {(\frac{x}{2})}_{x = π} = 0

Commented by mathlove last updated on 28/Oct/22

d i s c r a i b a n s w e r ? ?

Commented by mr W last updated on 29/Oct/22

\lim_{x \to π} \frac{s i n \frac{x}{2} - 1}{x - π}

= \lim_{x \to π} \frac{s i n \frac{x}{2} - \sin \frac{π}{2}}{x - π} \leftarrow d e r i v a t i v e o f \sin \frac{x}{2}

= {(\sin \frac{x}{2})}^{'} ∣_{x = π}

= (\frac{1}{2} \cos \frac{x}{2}) ∣_{x = π}

= \frac{1}{2} \cos \frac{π}{2}

= 0

Commented by mathlove last updated on 29/Oct/22

t a n k s

Commented by CElcedricjunior last updated on 29/Oct/22

\lim_{x \to π} \frac{\sin \frac{x}{2} - 1}{x - π} = \frac{0}{0} = FI

to apply hospital

\Leftrightarrow \lim_{x \to π} \frac{\sin \frac{x}{2} - 1}{x - π} = \lim_{x \to π} \frac{\frac{1}{2} \cos \frac{x}{2}}{1} = 0

\lim_{x \to π} \frac{\sin \frac{x}{2} - 1}{x - π} = 0

\dots \dots \dots . . l e c e l e b r e c e d r i c j i n i o r \dots \dots

Answered by a.lgnaoui last updated on 28/Oct/22

\lim_{x \to π} \frac{1}{2} \times (\frac{\sin \frac{x}{2} - \sin \frac{π}{2}}{\frac{x}{2} - \frac{π}{2}}) = \frac{1}{2} \times \frac{d {(\sin \frac{x}{2})}_{x = π / 2}}{d x} = \frac{1}{4} \cos (\frac{π}{4}) = \frac{1}{4} \times \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{8}

Commented by Acem last updated on 29/Oct/22

f i x {(\sin θ)}^{'}

Answered by cortano1 last updated on 29/Oct/22

\lim_{x \to π} \frac{\sin \frac{x}{2} - 1}{x - π}

let x - π = y \Rightarrow x = π + y

L = \lim_{y \to 0} \frac{\sin (\frac{π}{2} + \frac{y}{2}) - 1}{y}

= \lim_{y \to 0} \frac{\cos \frac{y}{2} - 1}{y}

= \lim_{y \to 0} \frac{- \sin^{2} (\frac{y}{2})}{(\cos \frac{y}{2} + 1) y}

= - \lim_{y \to 0} \frac{\sin (\frac{y}{2})}{y} . \lim_{y \to 0} \frac{\sin (\frac{y}{2})}{\cos (\frac{y}{2}) + 1}

= - \frac{1}{2} \times 0 = 0

Commented by mathlove last updated on 29/Oct/22

t h a n k s