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Factorization Method Form to Remove Indeterminate Form

Solve: lim x...

Question

Solve:
$lim x \to π / 6 \frac{2 sin x - 1}{4 {cos}^{2} x - 3}$

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Solution

${lim}_{x \to \frac{π}{6}} \frac{2 sin x - 1}{4 {cos}^{2} x - 3}$
$= {lim}_{x \to \frac{π}{6}} \frac{2 sin x - 1}{4 (1 - {sin}^{2} x) - 3}$
$= {lim}_{x \to \frac{π}{6}} \frac{2 sin x - 1}{4 - 4 {sin}^{2} x - 3}$
$= {lim}_{x \to \frac{π}{6}} \frac{2 sin x - 1}{1 - 4 {sin}^{2} x}$
$= {lim}_{x \to \frac{π}{6}} \frac{2 sin x - 1}{(1 - 2 sin x) (1 + 2 sin x)}$
$= {lim}_{x \to \frac{π}{6}} \frac{- 1}{(1 + 2 sin x)}$
$= \frac{- 1}{(1 + 2 sin \frac{π}{6})}$
$= \frac{- 1}{1 + 1 \times \frac{1}{2}}$
$= \frac{- 1}{1 + 1} = \frac{- 1}{2}$

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