Evaluate the limit- limx- 0sin 5x-sin 3xsin x

We have, lim_x→ 0sin 5x sin 3xsin x It becomes 0/0 form. nbsp;[∵ sin A sin B = 2 cosA+B2 sin A B2 =lim_x→ 02 cos(5x+3x2) sin(5x 3x2)sin x = lim_x→ 02 cos 4 ...

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Standard XII

Mathematics

Expansions to Remove Indeterminate Form

Evaluate the ...

Question

Evaluate the limit:

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

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Solution

We have,

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

It becomes 0/0 form.

$[∵ s i n A - s i n B = 2 c o s \frac{A + B}{2} s i n \frac{A - B}{2}$

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

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

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

$= 2 c o s (0)$

$= 2 \times 1$

$= 2$

Therefore,

$lim x \to 0 \frac{s i n 5 x - s i n 3 x}{s i n x} = 2$

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Expansions to Remove Indeterminate Form

Standard XII Mathematics

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Evaluate the limit: limx→0sin 5x−sin 3xsin x

We have, limx→0sin 5x−sin 3xsin x It becomes 0/0 form. [∵sin A−sin B=2 cosA+B2sinA−B2 =limx→02 cos(5x+3x2) sin(5x−3x2)sin x =limx→02 cos 4x sin xsin x =limx→02 cos 4x =2 cos(0) =2×1 =2 Therefore, limx→0sin 5x−sin 3xsinx=2

Evaluate the limit:

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