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$$\begin{gathered} \mathrm{We} \mathrm{have} \\ \mathrm{L}=\underset{\mathrm{x}\to 0}{\lim } \frac{\sin \left(-8\mathrm{x}\right)}{\tan 7\mathrm{x}} \\ =-\underset{\mathrm{x}\to 0}{\lim } \frac{\sin \left(8\mathrm{x}\right)}{\tan 7\mathrm{x}} \left[\mathrm{As} \sin \left(-\mathrm{A}\right)=-\mathrm{sinA}\right] \\ =-\underset{\mathrm{x}\to 0}{\lim } \frac{\sin \left(8\mathrm{x}\right)}{\mathrm{x}}.\frac{\mathrm{x}}{\tan \left(7\mathrm{x}\right)} \\ =-\frac{8}{7}\underset{\mathrm{x}\to 0}{\lim } \frac{\sin \left(8\mathrm{x}\right)}{\left(8\mathrm{x}\right)}.\frac{\left(7\mathrm{x}\right)}{\tan \left(7\mathrm{x}\right)} \\ =-\frac{8}{7}\underset{\mathrm{x}\to 0}{\lim } \frac{\sin \left(8\mathrm{x}\right)}{\left(8\mathrm{x}\right)}\underset{\mathrm{x}\to 0}{\lim }\frac{\left(7\mathrm{x}\right)}{\tan \left(7\mathrm{x}\right)} \\ =-\frac{8}{7}\times 1\times 1 \\ =-\frac{8}{7} \left(\mathrm{Option}\left(\mathrm{b}\right)\right) \\ \left(\mathrm{iii}\right) \mathrm{y}=\log \left({\mathrm{e}}^{\mathrm{x}}.{\mathrm{x}}^{\mathrm{n}}.3^{\mathrm{x}}\right) \\ \mathrm{y}=\log \left({\mathrm{e}}^{\mathrm{x}}\right)+\log \left({\mathrm{x}}^{\mathrm{n}}\right)+\log \left(3^{\mathrm{x}}\right) \\ \Rightarrow \mathrm{y}=\mathrm{xlog}\left(\mathrm{e}\right)+\mathrm{nlog}\left(\mathrm{x}\right)+\mathrm{xlog}\left(3\right) \\ \Rightarrow \mathrm{y}=\mathrm{x}+\mathrm{nlog}\left(\mathrm{x}\right)+\mathrm{xlog}\left(3\right) \left[\mathrm{As} \log \left(\mathrm{e}\right)=1\right] \\ \mathrm{Differentiate} \\ \frac{\mathrm{dy}}{\mathrm{dx}}=1+\frac{\mathrm{n}}{\mathrm{x}}+\log \left(3\right)..\left(\mathrm{Option}\left(\mathrm{b}\right)\right) \end{gathered}$$

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