Evaluate lim x - 0 cos 8 x - cos 2 x cos 12 x - cos 4 x

lim_x→ 0cos8x cos2xcos12x cos4x Using transformation angle formula, cosC cosD= 2sin(C+D2)sin(C D2) =lim_x→ 0 2sin(8x+2x2)sin(8x 2x2) 2sin(12x+4x2) ...

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Byju's Answer

Standard XII

Mathematics

Sandwich Theorem

Evaluate lim...

Question

Evaluate
${lim}_{x \to 0} \frac{cos 8 x - cos 2 x}{cos 12 x - cos 4 x}$

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Solution

${lim}_{x \to 0} \frac{cos 8 x - cos 2 x}{cos 12 x - cos 4 x}$
Using transformation angle formula, $cos C - cos D = - 2 sin (\frac{C + D}{2}) sin (\frac{C - D}{2})$

$= {lim}_{x \to 0} \frac{- 2 sin (\frac{8 x + 2 x}{2}) sin (\frac{8 x - 2 x}{2})}{- 2 sin (\frac{12 x + 4 x}{2}) sin (\frac{12 x - 4 x}{2})}$

$= {lim}_{x \to 0} \frac{sin (\frac{10 x}{2}) sin (\frac{6 x}{2})}{sin (\frac{16 x}{2}) sin (\frac{8 x}{2})}$

$= {lim}_{x \to 0} \frac{sin 5 x sin 3 x}{sin 8 x sin 4 x}$

$= {lim}_{x \to 0} \frac{\frac{sin 5 x}{5 x} \times 5 x \frac{sin 3 x}{3 x} \times 3 x}{\frac{sin 8 x}{8 x} \times 8 x \frac{sin 4 x}{4 x} \times 4 x}$

We know that,
${lim}_{x \to 0} \frac{sin a x}{a x} = 1$

$= {lim}_{x \to 0} \frac{5 x \times 3 x}{8 x \times 4 x}$

$= {lim}_{x \to 0} \frac{5 \times 3}{8 \times 4}$

$= {lim}_{x \to 0} \frac{15}{32} = 0.46875 \sim 0.47$

Suggest Corrections

Evaluatelimx→0cos8x−cos2xcos12x−cos4x

Evaluate
${lim}_{x \to 0} \frac{cos 8 x - cos 2 x}{cos 12 x - cos 4 x}$