lim-x-pi-3-cos-x-sin-pi-6-pi-6-x-2-

Question Number 55592 by gunawan last updated on 27/Feb/19

\lim_{x \to π / 3} \frac{\cos x - \sin \frac{π}{6}}{\frac{π}{6} - \frac{x}{2}} = . .

Commented by maxmathsup by imad last updated on 27/Feb/19

l e t A (x) = \frac{c o s x - s i n (\frac{π}{6})}{\frac{π}{6} - \frac{x}{2}} \Rightarrow A (x) = 6 \frac{c o s x - \frac{1}{2}}{π - 3 x} = 3 \frac{2 c o s x - 1}{3 (\frac{π}{3} - x)}

= \frac{1 - 2 c o s x}{x - \frac{π}{3}} c h a n g e m e n t x - \frac{π}{3} = t g i v e {l i m}_{x \to \frac{π}{3}} A (x)

= {l i m}_{t \to 0} \frac{1 - 2 c o s (t + \frac{π}{3})}{t} = {l i m}_{t \to 0} \frac{1 - 2 {\frac{1}{2} c o s t - \frac{\sqrt{3}}{2} s i n t)}}{t}

= {l i m}_{t \to 0} \frac{1 - c o s t + \sqrt{3} s i n t}{t} = {l i m}_{t \to 0} \frac{1 - c o s t}{t} + {l i m}_{t \to 0} \sqrt{3} \frac{s i n t}{t}

b u t c o s t \sim 1 - \frac{t^{2}}{2} \Rightarrow \frac{1 - c o s t}{t} \sim \frac{t}{2} \to 0 (t \to 0) a l s o s i n t \sim t \Rightarrow {l i m}_{t \to 0} \frac{s i n t}{t} = 1 \Rightarrow

{l i m}_{x \to \frac{π}{3}} A (x) = \sqrt{3} .

Answered by kaivan.ahmadi last updated on 27/Feb/19

h o p i t a l

l i \underset{x \to \frac{π}{3}}{m} \frac{- s i n x}{- \frac{1}{2}} = l i \underset{x \to \frac{π}{3}}{m} 2 s i n x = 2 s i n \frac{π}{3} = 2 \times \frac{\sqrt{3}}{2} = \sqrt{3}

Answered by math1967 last updated on 27/Feb/19

\lim_{x \to \frac{π}{3}} \frac{cosx - \cos (\frac{π}{2} - \frac{π}{6})}{\frac{π}{6} - \frac{x}{2}}

\lim_{x \to \frac{π}{3}} \frac{2 \sin (\frac{π}{6} - \frac{x}{2}) \sin (\frac{x}{2} + \frac{π}{6})}{\frac{π}{6} - \frac{x}{2}} ★

2 \times 1 \times \sin (\frac{π}{6} + \frac{π}{6}) = 2 \times 1 \times \frac{\sqrt{3}}{2} = \sqrt{3}

★ \lim_{(\frac{π}{6} - \frac{x}{2}) \to 0} \frac{\sin (\frac{π}{6} - \frac{x}{2})}{(\frac{π}{6} - \frac{x}{2})} = 1

∵ x \to \frac{π}{3} ∴ \frac{x}{2} \to \frac{π}{6} ∴ (\frac{π}{6} - \frac{x}{2}) \to 0

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