Lim x - 0sin 2-x-sin 2-x-x

Lim_x → 0sin(2+x) sin(2 x)/x =lim_x → 02cos ( 2+x+2 x/2 )sin (2+x 2+x/2 )/x=2lim_x → 0cos(2)× sinx/x =2lim_x → 0cos2×lim_x → 0sinx/x [ ∵lim_x → 0s ...

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Byju's Answer

Standard XII

Mathematics

Standard Limits to Remove Indeterminate Form

lim x → 0sin ...

Question

${lim}_{x \to 0} \frac{s i n (2 + x) - s i n (2 - x)}{x}$

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Solution

${lim}_{x \to 0} \frac{s i n (2 + x) - s i n (2 - x)}{x}$

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

$= 2 {lim}_{x \to 0} c o s 2 \times {lim}_{x \to 0} \frac{s i n x}{x} [∵ {lim}_{x \to 0} \frac{s i n x}{x} = 1]$

$= 2 c o s 2 \times 1 = 2 c o s 2$

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Similar questions

\lim_{x \to 0} \frac{\sin (2 + x) - \sin (2 - x)}{x}

Q. Prove that:

lim x \to 0 \frac{sin (2 + x) - sin (2 - x)}{x}

Q. Evaluate

lim n \to 0 \frac{{sin}^{2} (x + n) - {sin}^{2} x}{n}

Q. Solve:

lim x \to \frac{sin (2 + x) - sin (2 - x)}{x}

Q. Assertion :

lim x \to 0 {1^{1 / {sin}^{2} x} + 2^{1 / {sin}^{2} x} + 3^{1 / {sin}^{2} x} + . . . . . . . . + n^{1 / {sin}^{2} x}}^{{sin}^{2} x} = n

Reason: For

0 < a < 1, lim x \to \infty a^{x} = 0

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limx→0sin(2+x)−sin(2−x)x

limx→0sin(2+x)−sin(2−x)x =limx→02cos(2+x+2−x2)sin(2+x−2+x2)x=2limx→0cos(2)×sinxx =2limx→0cos2×limx→0sinxx [∵limx→0sinxx=1] =2 cos 2×1=2 cos 2

${lim}_{x \to 0} \frac{s i n (2 + x) - s i n (2 - x)}{x}$