Evaluate the limit: lim _{x rightarrow 0} fra | vedclass

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Question

(B) Using the trigonometric identities $\sin C - \sin D = 2 \cos \left(\frac{C+D}{2}\right) \sin \left(\frac{C-D}{2}\right)$ and $\cos C + \cos D = 2

Vedclass · Accepted Answer

$2$

Vedclass · Answer

(B) Using the trigonometric identities $\sin C - \sin D = 2 \cos \left(\frac{C+D}{2}ight) \sin \left(\frac{C-D}{2}ight)$ and $\cos C + \cos D = 2 \cos \left(\frac{C+D}{2}ight) \cos \left(\frac{C-D}{2}ight)$: Numerator: $4[\sin (2022 x) - \sin (2020 x)] = 4 	imes 2 \cos(2021 x) \sin(x) = 8 \cos(2021 x) \sin(x)$. Denominator: $x[\cos (2022 x) + \cos (2020 x) + 2 \cos (2021 x)] = x[2 \cos(2021 x) \cos(x) + 2 \cos(2021 x)] = 2x \cos(2021 x) [\cos(x) + 1]$. Substituting these into the limit: $\lim _{x ightarrow 0} \frac{8 \cos(2021 x) \sin(x)}{2x \cos(2021 x) [\cos(x) + 1]} = \lim _{x ightarrow 0} \frac{4 \sin(x)}{x [\cos(x) + 1]}$. Since $\lim _{x ightarrow 0} \frac{\sin x}{x} = 1$ and $\cos(0) = 1$: $= \frac{4(1)}{1 + 1} = \frac{4}{2} = 2$.

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{4[\sin (2022 x)-\sin (2020 x)]}{x[\cos (2022 x)+2 \cos (2021 x)+\cos (2020 x)]}$

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