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Question

Evaluate: $$\displaystyle \lim_{x\rightarrow 0}\frac{\sin 7x-\sin 5x+\sin 3x-\sin x}{\sin 6x-\sin 4x+\sin 2x}$$


A
-1
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B
0
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C
1
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D
2
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Solution

The correct option is A 1
$$\displaystyle \lim_{x\rightarrow 0}\dfrac{\sin 7x-\sin 5x+\sin 3x-\sin x}{\sin 6x-\sin 4x+\sin 2x}$$

$$=\displaystyle \lim_{x\rightarrow 0}\dfrac{7x\dfrac{\sin 7x}{7x}-\dfrac{5x\sin 5x}{5x}+\dfrac{3x\sin 3x}{3x}-\dfrac{x\sin x}{x}}{\dfrac{6x\sin 6x}{6x}-\dfrac{4x \sin 4x}{4x}+\dfrac{2x \sin 2x}{2x}}$$.....As$$\left [\displaystyle \lim_{x\rightarrow 0} \dfrac{sinx}x=1 \right ]$$

$$=\displaystyle \lim_{x\rightarrow 0}\dfrac{10x-6x}{8x-4x}$$

$$=\displaystyle \lim_{x\rightarrow 0}\,1$$

$$=1$$


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