Solve - x -6lim 3 sin x - -3cos x 6x - -

The correct option is B 1√(3) lim _ x→π #160; 6 #160; ( 3sin x √( 3 )cos x #160; 6x π #160; #160; ) #160; ⇒lim _ x→π #160; ...

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Byju's Answer

Standard XII

Mathematics

Substitution Method to Remove Indeterminate Form

Solve : x →π...

Question

Solve :
$lim x \to \frac{π}{6}$ $(\frac{3 sin x - \sqrt{3} cos x}{6 x - π})$ =

\sqrt{3}

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\frac{1}{\sqrt{3}}

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- \frac{1}{\sqrt{3}}

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\frac{2}{\sqrt{3}}

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Solution

The correct option is B $\frac{1}{\sqrt{3}}$
$lim x \to \frac{π}{6} (\frac{3 sin x - \sqrt{3} cos x}{6 x - π})$

$\Rightarrow lim x \to \frac{π}{6} \frac{\sqrt{3} (\sqrt{3} sin x - cos x)}{6 (x - \frac{π}{6})}$

$\Rightarrow lim (x \to \frac{π}{6}) ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \frac{(2 \sqrt{3}) [\frac{\sqrt{3}}{2} sin x - \frac{1}{2} cos x]}{6 (x - \frac{π}{6})} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦$

$\Rightarrow lim (x \to \frac{π}{6}) \frac{1}{\sqrt{3}} ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \frac{(cos (\frac{π}{6}) sin x - sin \frac{π}{6} cos x)}{(x - \frac{π}{6})} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦$

$\Rightarrow lim (x \to \frac{π}{6}) \frac{1}{\sqrt{3}} ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \frac{sin (x - \frac{π}{6})}{(x - \frac{π}{6})} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦$

$\Rightarrow \frac{1}{\sqrt{3}} lim (x \to \frac{π}{6}) ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \frac{sin (x - \frac{π}{6})}{(x - \frac{π}{6})} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦$

$\Rightarrow \frac{1}{\sqrt{3}} (1) \Rightarrow \frac{1}{\sqrt{3}}$

$∴ lim x \to \frac{π}{6} \frac{(3 sin x - \sqrt{3} cos x)}{(6 x - π)} = \frac{1}{\sqrt{3}}$

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Solve :limx→π6 (3sinx−√3cosx6x−π)=

Solve :
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