Question

Lim x tends to 2024 (sinx-sin2024) /(cosx-cos2024)

Answer 1

-cot(2024)

Answer 2

The limit is of the form $\frac{0}{0}$ as $x \to 2024$. Using trigonometric identities $\sin x - \sin a = 2 \cos\left(\frac{x+a}{2}\right) \sin\left(\frac{x-a}{2}\right)$ and $\cos x - \cos a = -2 \sin\left(\frac{x+a}{2}\right) \sin\left(\frac{x-a}{2}\right)$, the expression simplifies to $-\cot\left(\frac{x+a}{2}\right)$ for $x \neq a$. Taking the limit as $x \to a=2024$, we get $-\cot\left(\frac{a+a}{2}\right) = -\cot(a) = -\cot(2024)$. Alternatively, applying L'Hopital's rule, the limit is $\lim_{x \to 2024} \frac{\frac{d}{dx}(\sin x - \sin 2024)}{\frac{d}{dx}(\cos x - \cos 2024)} = \lim_{x \to 2024} \frac{\cos x}{-\sin x} = \frac{\cos 2024}{-\sin 2024} = -\cot 2024$.