The value oflimx-01-cos3x-xsin xcosx

Lim_x→01 cos^3x/xsin xcosx =lim_x→0(1 cos x)(1+cos x+cos^2x)/xsin xcos x =lim_x→02sin^2 ( x/2 )/2sin ( x/2 )cos ( x/2 ). x×(1+cosx+cos^2x)/cos x =lim_x→0sin ...

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Standard XIII

Mathematics

Factorization Method Form to Remove Indeterminate Form

The value ofl...

Question

$The value of {lim}_{x \to 0} \frac{1 - c o s^{3} x}{x s i n x c o s x}$

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Solution

The correct option is C
3/2

${lim}_{x \to 0} \frac{1 - c o s^{3} x}{x s i n x c o s x} = {lim}_{x \to 0} \frac{(1 - c o s x) (1 + c o s x + c o s^{2} x)}{x s i n x c o s x} = {lim}_{x \to 0} \frac{2 s i n^{2} (\frac{x}{2})}{2 s i n (\frac{x}{2}) c o s (\frac{x}{2}) . x} \times \frac{(1 + c o s x + c o s^{2} x)}{c o s x} = {lim}_{x \to 0} \frac{s i n (\frac{x}{2})}{2 (\frac{x}{2})} \times \frac{1 + c o s x + c o s^{2} x}{c o s (\frac{x}{2}) c o s x} = \frac{1}{2} \times 3 = \frac{3}{2}$

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The value oflimx→01−cos3xxsin xcosx

The correct option is C 3/2 limx→01−cos3xxsin xcosx=limx→0(1−cos x)(1+cos x+cos2x)xsin xcos x=limx→02sin2(x2)2sin(x2)cos(x2).x×(1+cosx+cos2x)cos x=limx→0sin(x2)2(x2)×1+cosx+cos2xcos(x2)cos x=12×3=32

$The value of {lim}_{x \to 0} \frac{1 - c o s^{3} x}{x s i n x c o s x}$