Prove that- lim x - 0 sin 2 - x - sin 2 - x - x

x→ 0ltsin(2+x) sin(2 x)x =x→ 0lt2cos(2+x+2 x2)sin(2+x 2+x2)x =x→ 0lt 2cos 2sin xx =2cos 2·x→ 0ltsin xx =2cos 2 . ...

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Greatest Integer Function

Prove that: ...

Question

Prove that:
$lim x \to 0 \frac{sin (2 + x) - sin (2 - x)}{x}$

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

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

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

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

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

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