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Question Number 201680 by cortano12 last updated on 10/Dec/23

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Answered by Calculusboy last updated on 11/Dec/23

$$\mathit{S}\mathit{o}\mathit{l}\mathit{u}\mathit{t}\mathit{i}\mathit{o}\mathit{n}:\mathit{s}\mathit{u}\mathit{b}\mathit{s}\mathit{t}\mathit{i}\mathit{t}\mathit{u}\mathit{t}\mathit{e}\mathit{d}\mathit{i}\mathit{t}\mathit{e}\mathit{c}\mathit{t}\mathit{l}\mathit{y},\mathit{w}\mathit{e}\mathit{g}\mathit{e}\mathit{t}\frac{0}{0}\left(\mathit{i}\mathit{n}\mathit{d}\mathit{e}\mathit{t}\mathit{e}\mathit{r}\mathit{m}\mathit{i}\mathit{n}\mathit{a}\mathit{n}\mathit{t}\right)$$$$\mathit{l}\mathit{e}\mathit{t}\mathit{p}=2\mathit{s}\mathit{i}\mathit{n}\mathit{x}-\mathit{s}\mathit{i}\mathit{m}2\mathit{x}\frac{\mathit{d}\mathit{p}}{\mathit{d}\mathit{x}}=2\mathit{c}\mathit{o}\mathit{s}\mathit{x}-2\mathit{c}\mathit{o}\mathit{s}2\mathit{x}$$$$\mathit{l}\mathit{e}\mathit{t}\mathit{q}=\mathit{s}\mathit{i}\mathit{n}\mathit{x}-\mathit{x}\mathit{c}\mathit{o}\mathit{s}\mathit{x}\frac{\mathit{d}\mathit{q}}{\mathit{d}\mathit{x}}=\mathit{c}\mathit{o}\mathit{s}\mathit{x}-(\mathit{c}\mathit{o}\mathit{s}\mathit{x}-\mathit{x}\mathit{s}\mathit{i}\mathit{n}\mathit{x})$$$$\mathit{l}\mathit{i}\underset{\mathit{x}\to 0}{\mathit{m}}\frac{\mathit{p}}{\mathit{q}}=\mathit{l}\mathit{i}\underset{\mathit{x}\to 0}{\mathit{m}}\frac{2\mathit{c}\mathit{o}\mathit{s}\mathit{x}-2\mathit{c}\mathit{o}\mathit{s}\left(2\mathit{x}\right)}{\mathit{c}\mathit{o}\mathit{s}\mathit{x}-\mathit{c}\mathit{o}\mathit{s}\mathit{x}+\mathit{x}\mathit{s}\mathit{i}\mathit{n}\mathit{x}}=\mathit{l}\mathit{i}\underset{\mathit{x}\to 0}{\mathit{m}}\frac{2\mathit{c}\mathit{o}\mathit{s}\mathit{x}-2\mathit{c}\mathit{o}\mathit{s}\left(2\mathit{x}\right)}{\mathit{x}\mathit{s}\mathit{i}\mathit{n}\mathit{x}}$$$$\frac{\mathit{l}\mathit{i}\underset{\mathit{x}\to 0}{\mathit{m}}(2\mathit{c}\mathit{o}\mathit{s}\mathit{x}-2\mathit{c}\mathit{o}\mathit{s}\left(2\mathit{x}\right))}{\mathit{l}\mathit{i}\underset{\mathit{x}\to 0}{\mathit{m}}\left(\mathit{x}\mathit{s}\mathit{i}\mathit{n}\mathit{x}\right)}=\frac{0}{0}$$$$\mathit{l}\mathit{e}\mathit{t}\mathit{u}=2\mathit{c}\mathit{o}\mathit{s}\mathit{x}-2\mathit{c}\mathit{o}\mathit{s}\left(2\mathit{x}\right)\frac{\mathit{d}\mathit{u}}{\mathit{d}\mathit{x}}=-2\mathit{s}\mathit{i}\mathit{n}\mathit{x}+4\mathit{s}\mathit{i}\mathit{n}\left(2\mathit{x}\right)$$$$\mathit{l}\mathit{e}\mathit{t}\mathit{k}=\mathit{x}\mathit{s}\mathit{i}\mathit{n}\mathit{x}\frac{\mathit{d}\mathit{k}}{\mathit{d}\mathit{x}}=\mathit{s}\mathit{i}\mathit{n}\mathit{x}+\mathit{x}\mathit{c}\mathit{o}\mathit{s}\mathit{x}$$$$\mathit{l}\mathit{i}\underset{\mathit{x}\to 0}{\mathit{m}}\frac{\mathit{u}}{\mathit{k}}=\mathit{l}\mathit{i}\underset{\mathit{x}\to 0}{\mathit{m}}\frac{4\mathit{s}\mathit{i}\mathit{n}\left(2\mathit{x}\right)-2\mathit{s}\mathit{i}\mathit{n}\mathit{x}}{\mathit{s}\mathit{i}\mathit{n}\mathit{x}+\mathit{x}\mathit{c}\mathit{o}\mathit{s}\mathit{x}}=\frac{0}{0}$$$$\mathit{l}\mathit{e}\mathit{t}\mathit{j}=4\mathit{s}\mathit{i}\mathit{n}\left(2\mathit{x}\right)-2\mathit{s}\mathit{i}\mathit{n}\mathit{x}\frac{\mathit{d}\mathit{j}}{\mathit{d}\mathit{x}}=8\mathit{c}\mathit{o}\mathit{s}\left(2\mathit{x}\right)-2\mathit{c}\mathit{o}\mathit{s}\mathit{x}$$$$\mathit{l}\mathit{e}\mathit{t}\mathit{g}=\mathit{s}\mathit{i}\mathit{n}\mathit{x}+\mathit{x}\mathit{c}\mathit{o}\mathit{s}\mathit{x}\frac{\mathit{d}\mathit{g}}{\mathit{d}\mathit{x}}=\mathit{c}\mathit{o}\mathit{s}\mathit{x}+(\mathit{c}\mathit{o}\mathit{s}\mathit{x}-\mathit{x}\mathit{s}\mathit{i}\mathit{n}\mathit{c})=2\mathit{c}\mathit{o}\mathit{s}\mathit{x}-\mathit{x}\mathit{s}\mathit{i}\mathit{n}\mathit{x}$$$$\mathit{l}\mathit{i}\underset{\mathit{x}\to 0}{\mathit{m}}\frac{\mathit{j}}{\mathit{g}}=\mathit{l}\mathit{i}\underset{\mathit{x}\to 0}{\mathit{m}}\frac{8\mathit{c}\mathit{o}\mathit{s}\left(2\mathit{x}\right)-2\mathit{c}\mathit{o}\mathit{s}\mathit{x}}{2\mathit{c}\mathit{o}\mathit{s}\mathit{x}-\mathit{x}\mathit{s}\mathit{i}\mathit{n}\mathit{x}}=\frac{8-2}{2-0}=\frac{6}{2}=3$$$$\therefore \mathit{l}\mathit{i}\underset{\mathit{x}\to 0}{\mathit{m}}\frac{2\mathit{s}\mathit{i}\mathit{n}\mathit{x}-\mathit{s}\mathit{i}\mathit{n}\left(2\mathit{x}\right)}{\mathit{s}\mathit{i}\mathit{n}\mathit{x}-\mathit{x}\mathit{c}\mathit{o}\mathit{s}\mathit{x}}=3$$$$$$

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