Answer
$$\frac{4}{3}$$
Work Step by Step
\[\begin{gathered}
\mathop {\lim }\limits_{x \to 0} \frac{{\sin 4x}}{{\sin 3x}} \hfill \\
{\text{Evaluate }}f\left( x \right){\text{ for the given values and complete the table}}{\text{.}} \hfill \\
x = - 0.1 \to f\left( { - 0.1} \right) = \frac{{\sin \left[ {4\left( { - 0.1} \right)} \right]}}{{\sin \left[ {3\left( { - 0.1} \right)} \right]}} \approx 1.3177 \hfill \\
x = - 0.01 \to f\left( { - 0.01} \right) = \frac{{\sin \left[ {4\left( { - 0.01} \right)} \right]}}{{\sin \left[ {3\left( { - 0.01} \right)} \right]}} \approx 1.3332 \hfill \\
x = - 0.001 \to f\left( { - 0.001} \right) = \frac{{\sin \left[ {4\left( { - 0.001} \right)} \right]}}{{\sin \left[ {3\left( { - 0.001} \right)} \right]}} \approx 1.3333 \hfill \\
x = 0.001 \to f\left( {0.001} \right) = \frac{{\sin \left[ {4\left( {0.001} \right)} \right]}}{{\sin \left[ {3\left( {0.001} \right)} \right]}} \approx 1.3333 \hfill \\
x = 0.01 \to f\left( {0.01} \right) = \frac{{\sin \left[ {4\left( {0.01} \right)} \right]}}{{\sin \left[ {3\left( {0.01} \right)} \right]}} \approx 1.3332 \hfill \\
x = 0.1 \to f\left( {0.1} \right) = \frac{{\sin \left[ {4\left( {0.1} \right)} \right]}}{{\sin \left[ {3\left( {0.1} \right)} \right]}} \approx 1.3177 \hfill \\
\boxed{\begin{array}{*{20}{c}}
x&{ - 0.1}&{ - 0.01}&{ - 0.001}&{0.001}&{0.01}&{0.1} \\
{f\left( x \right)}&{1.3177}&{1.3332}&{1.3333}&{1.3333}&{1.3332}&{1.3177}
\end{array}} \hfill \\
{\text{Therefore,}} \hfill \\
\mathop {\lim }\limits_{x \to 0} \frac{{\sin 4x}}{{\sin 3x}} \approx 1.3333 \hfill \\
{\text{Exact value :}} \hfill \\
\mathop {\lim }\limits_{x \to 0} \frac{{\sin 4x}}{{\sin 3x}} = \frac{4}{3} \hfill \\
{\text{Graph}} \hfill \\
\end{gathered} \]