Lim x - 0sin 5 x-sin 3 x-sin x

Lim_x→ 0sin5x sin3x/sinx =lim_x→ 02cos ( 5x+3x/2 )sin ( 5x 3x/2 )/sinx [Applying Componendo amp; dividendo formula] =lim_x→ 0 ( 2cos4xsinx/sinx ) =2 lim_x→ 0 ...

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Byju's Answer

Standard XII

Mathematics

Standard Limits to Remove Indeterminate Form

lim x → 0sin ...

Question

${lim}_{x \to 0} \frac{s i n 5 x - s i n 3 x}{s i n x}$

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Solution

${lim}_{x \to 0} \frac{s i n 5 x - s i n 3 x}{s i n x}$

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

[Applying Componendo & dividendo formula]

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

$= 2 {lim}_{x \to 0} c o s 4 x$

$= 2 \times c o s 0 = 2$

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lim x \to 0 \frac{sin 5 x - sin 3 x}{sin x}

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{lim}_{x \to 0} \frac{tan x - sin x}{{sin}^{3} x}

${lim}_{x \to 0} \frac{s i n 5 x}{t a n 3 x}$

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limx→0sin5x−sin3xsinx

limx→0sin5x−sin3xsinx =limx→02cos(5x+3x2)sin(5x−3x2)sinx [Applying Componendo & dividendo formula] =limx→0(2cos4xsinxsinx) =2 limx→0cos 4x =2×cos 0=2

${lim}_{x \to 0} \frac{s i n 5 x - s i n 3 x}{s i n x}$