Lim x - 01-cos 2 x-tan 2 x-x sin x

Lim_x→ 01 cos2x+tan^2x/xsinx =lim_x→ 02sin^2x+tan^2x/xsinx =2lim_x→ 0sin^2x+lim_x→ 0tan^2x/lim_x→ 0xsinx= ( 2 ( lim_x→ 0sinx/x )^2 × x^2 )+ (lim_x→ 0tanx/x )^2 ...

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Byju's Answer

Standard XII

Mathematics

Standard Limits to Remove Indeterminate Form

lim x → 01-co...

Question

${lim}_{x \to 0} \frac{1 - c o s 2 x + t a n^{2} x}{x s i n x}$

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Solution

${lim}_{x \to 0} \frac{1 - c o s 2 x + t a n^{2} x}{x s i n x}$

$= {lim}_{x \to 0} \frac{2 s i n^{2} x + t a n^{2} x}{x s i n x}$

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

$= \frac{(2 \times 1 \times x^{2}) + (1 \times x^{2})}{(1 \times x^{2})} [∵ {lim}_{x \to 0} \frac{s i n x}{x} = 1 and {lim}_{x \to 0} \frac{t a n x}{x} = 1]$

$= \frac{3 x^{2}}{x^{2}} = 3$

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limx→01−cos2x+tan2xxsinx

limx→01−cos2x+tan2xxsinx =limx→02sin2x+tan2xxsinx =2limx→0sin2x+limx→0tan2xlimx→0xsinx=(2(limx→0sinxx)2×x2)+(limx→0tanxx)2×x2limx→0sinxx×x2 =(2×1×x2)+(1×x2)(1×x2) [∵limx→0sinxx=1 andlimx→0tanxx=1] =3x2x2=3

${lim}_{x \to 0} \frac{1 - c o s 2 x + t a n^{2} x}{x s i n x}$