examine the continuity of the function
f(x)= sin 2x/ sin 3x , x is not 0
                 2 , x=0                 at x=0

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$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right)=\left\{\begin{array}{ll}\frac{\sin \left(2\mathrm{x}\right)}{\sin \left(3\mathrm{x}\right)}& \mathrm{x}\ne 0\\ 2& \mathrm{x}=0\end{array}\right. \\ \mathrm{f}\left(0\right)=2 \\ \underset{\mathrm{x}\to 0}{\lim } \mathrm{f}\left(\mathrm{x}\right)=\underset{\mathrm{x}\to 0}{\lim }\frac{\sin \left(2\mathrm{x}\right)}{\sin \left(3\mathrm{x}\right)} \\ =\underset{\mathrm{x}\to 0}{\lim }\frac{\left(2\mathrm{x}\right){\displaystyle \frac{\sin \left(2\mathrm{x}\right)}{\left(2\mathrm{x}\right)}}}{\left(3\mathrm{x}\right){\displaystyle \frac{\sin \left(3\mathrm{x}\right)}{\left(3\mathrm{x}\right)}}} \\ \mathrm{We} \mathrm{know} \mathrm{that} \underset{\mathrm{x}\to \mathrm{a}}{\lim } \frac{\mathrm{sinf}\left(\mathrm{x}\right)}{\mathrm{f}\left(\mathrm{x}\right)}=1 \mathrm{when} \underset{\mathrm{x}\to \mathrm{a}}{\lim } \mathrm{f}\left(\mathrm{x}\right)=0 \\ \underset{\mathrm{x}\to 0}{\lim } \mathrm{f}\left(\mathrm{x}\right) \\ =\underset{\mathrm{x}\to 0}{\lim }\frac{\left(2\mathrm{x}\right){\displaystyle \frac{\sin \left(2\mathrm{x}\right)}{\left(2\mathrm{x}\right)}}}{\left(3\mathrm{x}\right){\displaystyle \frac{\sin \left(3\mathrm{x}\right)}{\left(3\mathrm{x}\right)}}}=\underset{\mathrm{x}\to 0}{\lim } \frac{2\mathrm{x}}{3\mathrm{x}}.\frac{1}{1} \\ =\underset{\mathrm{x}\to 0}{\lim } \frac{2}{3} \\ \underset{\mathrm{x}\to 0}{\lim }\mathrm{f}\left(\mathrm{x}\right)=\frac{2}{3} \\ \mathrm{As} \underset{\mathrm{x}\to 0}{\lim }\mathrm{f}\left(\mathrm{x}\right) \ne \mathrm{f}\left(0\right) \\ \mathrm{Hence} \mathrm{function} \mathrm{is} \mathrm{discontinuous} \mathrm{at} \mathrm{x}=0 \end{gathered}$$

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