Lim x - 01-cos 2 x-cos 2 x-cos 8 x

Lim_x→ 01 cos2x/cos2x cos8x =lim_x→ 02sin^2x/ 2sin ( 2x+8x/2 )sin ( 2x 8x/2 )=lim_x→ 0sin^2x/sin5x sin( 3x) [Applying Componendo and dividendo formula] =lim_x→ ...

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Byju's Answer

Standard XII

Mathematics

Standard Limits to Remove Indeterminate Form

lim x → 01-co...

Question

${lim}_{x \to 0} \frac{1 - c o s 2 x}{c o s 2 x - c o s 8 x}$

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Solution

${lim}_{x \to 0} \frac{1 - c o s 2 x}{c o s 2 x - c o s 8 x}$

$= {lim}_{x \to 0} \frac{2 s i n^{2} x}{- 2 s i n (\frac{2 x + 8 x}{2}) s i n (\frac{2 x - 8 x}{2})} = {lim}_{x \to 0} \frac{s i n^{2} x}{s i n 5 x s i n (- 3 x)}$

[Applying Componendo and dividendo formula]

$= \frac{{lim}_{x \to 0} s i n^{2} x}{({lim}_{x \to 0} s i n 5 x) ({lim}_{x \to 0} s i n 3 x)} = \frac{{({lim}_{x \to 0} \frac{s i n 5 x}{x})}_{x \to 0}^{2} \times x^{2}}{({lim}_{x \to 0} \frac{s i n 5 x}{5 x}) \times 5 x ({lim}_{x \to 0} \frac{s i n 3 x}{3 x}) \times 3 x}$

$= \frac{1 \times x^{2}}{1 \times 5 x \times 1 \times 3 x} [∵ {lim}_{x \to 0} \frac{s i n x}{x} = 1]$

$= \frac{x^{2}}{15 x^{2}} = \frac{1}{15}$

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limx→01−cos2xcos2x−cos8x

${lim}_{x \to 0} \frac{1 - c o s 2 x}{c o s 2 x - c o s 8 x}$