Evaluate lim x - 0sin 2 x-sin 6 x-sin 5 x-sin 3 x

X → 0lim(sin 2x+sin 6x/sin 5x sin 3x ) =x → 0lim(2 sin 4x cos 2x/2 cos 4x sin x ) =x → 0lim(sin 4x cos 2x/cos 4x sin x ) =x → 0lim{sin 4x/4x×x/sin x× cos ...

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Byju's Answer

Standard XII

Mathematics

Differentiation of a Determinant

Evaluate lim ...

Question

Evaluate $l i m x \to 0 (\frac{s i n 2 x + s i n 6 x}{s i n 5 x - s i n 3 x})$

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Solution

$l i m x \to 0 (\frac{s i n 2 x + s i n 6 x}{s i n 5 x - s i n 3 x})$

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

$= l i m x \to 0 (\frac{s i n 4 x c o s 2 x}{c o s 4 x - s i n x})$

$= l i m x \to 0 {\frac{s i n 4 x}{4 x} \times \frac{x}{s i n x} \times c o s 2 x \times \frac{1}{c o s 4 x} \times 4}$

$= 4 \times l i m x \to 0 \frac{s i n 4 x}{4 x} \times l i m 4 x \to 0 (\frac{x}{s i n x}) \times l i m 2 x \to 0 c o s 2 x \times \frac{1}{l i m 4 x \to 0 c o s 4 x}$

$= (4 \times 1 \times 1 \times 1 \times \frac{1}{1}) = 4$

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then which of the following hold good?

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Evaluate limx→0(sin 2x+sin 6xsin 5x−sin 3x)

limx→0(sin 2x+sin 6xsin 5x−sin 3x) =limx→0(2 sin 4x cos 2x2 cos 4x sin x) =limx→0(sin 4x cos 2xcos 4x−sin x) =limx→0{sin 4x4x×xsin x×cos 2x×1cos 4x×4} =4×limx→0sin 4x4x×lim4x→0(xsin x)×lim2x→0 cos 2x×1lim4x→0 cos4x =(4×1×1×1×11)=4

Evaluate $l i m x \to 0 (\frac{s i n 2 x + s i n 6 x}{s i n 5 x - s i n 3 x})$