lim-x-pi-6-1-2sin-x-1-3-tan-x-

Question Number 117412 by bobhans last updated on 11/Oct/20

\lim_{x \to \frac{π}{6}} \frac{1 - 2 \sin x}{1 - \sqrt{3} \tan x} = ?

Commented by Lordose last updated on 11/Oct/20

you edited the question . ?

Commented by bobhans last updated on 11/Oct/20

yes . typo

Answered by Lordose last updated on 11/Oct/20

\frac{1 - 2 (\frac{1}{2})}{\sqrt{3} - \frac{\sqrt{3}}{3}} = \frac{0}{\frac{2 \sqrt{3}}{3}} = 0

Answered by Olaf last updated on 11/Oct/20

\lim_{x \to \frac{π}{6}} \frac{- 2 \cos x}{- \sqrt{3} (1 + \tan^{2} x)} = \frac{- 2 \frac{\sqrt{3}}{2}}{- \sqrt{3} (1 + {(\frac{1}{\sqrt{3}})}^{2})}

= \frac{3}{4}

({Hospital}^{'} s rule)

Answered by bemath last updated on 11/Oct/20

\lim_{x \to \frac{π}{6}} \frac{1 - 2 \sin x}{1 - \sqrt{3} \tan x} = ?

Solution :

Without L^{'} Hopital

letting w = x - \frac{π}{6}

\lim_{w \to 0} \frac{1 - 2 \sin (w + \frac{π}{6})}{1 - \sqrt{3} \tan (w + \frac{π}{6})} =

\lim_{w \to 0} \frac{1 - 2 (\frac{\sqrt{3}}{2} \sin w + \frac{1}{2} \cos w)}{1 - \sqrt{3} (\frac{\frac{1}{\sqrt{3}} + \tan w}{1 - \frac{1}{\sqrt{3}} \tan w})} =

\lim_{w \to 0} \frac{1 - \sqrt{3} \sin w - \cos w}{1 - \sqrt{3} (\frac{1 + \sqrt{3} \tan w}{\sqrt{3} - \tan w})} =

\lim_{w \to 0} \frac{(\sqrt{3} - \tan w) (1 - \cos w - \sqrt{3} \sin w)}{\sqrt{3} - \tan w - \sqrt{3} - 3 \tan w} =

\sqrt{3} \lim_{w \to 0} \frac{2 \sin^{2} (\frac{1}{2} w) - \sqrt{3} \sin w}{- 4 \tan w} =

\sqrt{3} \lim_{w \to 0} \frac{2 \sin (\frac{1}{2} w) (\sin (\frac{1}{2} w) - \sqrt{3} \cos (\frac{1}{2} w))}{- 4 \tan w} =

\frac{\sqrt{3}}{- 4} \times (- \sqrt{3}) = \frac{3}{4}

Commented by TANMAY PANACEA last updated on 11/Oct/20

\underset{x \to \frac{π}{6} \frac{}{}}{li}

\lim_{x \to \frac{π}{6}} \frac{2}{\sqrt{3}} (\frac{s i n \frac{π}{6} - s i n x}{t a n \frac{π}{6} - t a n x})

\frac{2}{\sqrt{3}} \lim_{x \to \frac{π}{6}} \frac{2 c o s (\frac{\frac{π}{6} + x}{2}) s i n (\frac{\frac{π}{6} - x}{2})}{\frac{s i n (\frac{π}{6} - x)}{c o s (\frac{π}{6}) c o s x}}

\frac{2}{\sqrt{3}} \lim_{x \to \frac{π}{6}} 2 c o s \frac{π}{6} c o s x \times c o s (\frac{\frac{π}{6} + x}{2}) s i n (\frac{\frac{π}{6} - x}{2}) \times \frac{1}{2 s i n (\frac{\frac{π}{6} - x}{2}) \times c o z (\frac{\frac{π}{6} - x}{2})}

= \frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{2} \times \frac{\sqrt{3}}{2} \times \frac{\sqrt{3}}{2} \times \frac{1}{1} = \frac{3}{4}