Lim x - acos x-cos 2-x-a

Lim_x → acos x cos a/x a =lim_x → a( 2 sin(x+a/2)sin (x a/2))/x a = 2 lim_x → a sin ( x+a/2 )lim_x → asin (x a/2)/x a = 2 × sin ( a+a/2 )× ( lim_x a → 0sin (x ...

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Byju's Answer

Standard XII

Mathematics

Standard Limits to Remove Indeterminate Form

lim x → acos ...

Question

${lim}_{x \to a} \frac{c o s x - c o s a}{x - a}$

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Solution

${lim}_{x \to a} \frac{c o s x - c o s a}{x - a}$

$= {lim}_{x \to a} \frac{(- 2 s i n (\frac{x + a}{2}) s i n (\frac{x - a}{2}))}{x - a}$

$= - 2 {lim}_{x \to a} s i n (\frac{x + a}{2}) {lim}_{x \to a} \frac{s i n (\frac{x - a}{2})}{x - a}$

$= - 2 \times s i n (\frac{a + a}{2}) \times ({lim}_{x - a \to 0} \frac{s i n (\frac{x - a}{2})}{\frac{x - a}{2}}) \times \frac{1}{2}$

$= - 2 s i n a \times 1 \times \frac{1}{2} [∵ {lim}_{x \to a} \frac{s i n x}{x} = 1]$

=-sin a

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limx→acos x−cos ax−a

limx→acos x−cos ax−a =limx→a(−2 sin(x+a2)sin(x−a2))x−a =−2limx→asin(x+a2)limx→asin(x−a2)x−a =−2×sin(a+a2)×(limx−a→0sin(x−a2)x−a2)×12 =−2 sin a×1×12[∵limx→asin xx=1] =-sin a

${lim}_{x \to a} \frac{c o s x - c o s a}{x - a}$