Lim x - 0tan x x-sin x-sin 3 x-sin x

Lim_x→ 0tanx sinx/sin3x 3sinx =lim_x→ 0sinx/cosx sinx/3sinx 4sin^3x 3sinx=lim_x→ 0sinx ( 1/cosx 1 )/ 4sin^3x= 1/4lim_x→ 01/cosx 1/sin^2x = 1/4lim_x→ 01 cosx/co ...

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Byju's Answer

Standard XII

Mathematics

Standard Limits to Remove Indeterminate Form

lim x → 0tan ...

Question

${lim}_{x \to 0} \frac{t a n x - s i n x}{s i n 3 x - 3 s i n x}$

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Solution

${lim}_{x \to 0} \frac{t a n x - s i n x}{s i n 3 x - 3 s i n x}$

$= {lim}_{x \to 0} \frac{\frac{s i n x}{c o s x} - s i n x}{3 s i n x - 4 s i n^{3} x - 3 s i n x} = {lim}_{x \to 0} \frac{s i n x (\frac{1}{c o s x} - 1)}{- 4 s i n^{3} x} = \frac{- 1}{4} {lim}_{x \to 0} \frac{\frac{1}{c o s x} - 1}{s i n^{2} x}$

$= \frac{- 1}{4} {lim}_{x \to 0} \frac{\frac{1 - c o s x}{c o s x}}{1 - c o s^{2} x} = \frac{- 1}{4} {lim}_{x \to 0} \frac{1 - c o s x}{(c o s x) (1 - c o s x) (1 + c o s x)}$

$= \frac{- 1}{4} {lim}_{x \to 0} \frac{1}{(c o s x) (1 + c o s x)} = \frac{- 1}{4} \times \frac{1}{1 (1 + 1)} = \frac{- 1}{4} \times \frac{1}{2} = - \frac{1}{8}$

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limx→0tanx−sinxsin3x−3sinx

limx→0tanx−sinxsin3x−3sinx =limx→0sinxcosx−sinx3sinx−4sin3x−3sinx=limx→0sinx(1cosx−1)−4sin3x=−14limx→01cosx−1sin2x =−14limx→01−cosxcosx1−cos2x=−14limx→01−cosx(cosx)(1−cosx)(1+cosx) =−14limx→01(cosx)(1+cosx)=−14×11(1+1)=−14×12=−18

${lim}_{x \to 0} \frac{t a n x - s i n x}{s i n 3 x - 3 s i n x}$