Answer
$3$
Work Step by Step
$$\eqalign{
& \mathop {\lim }\limits_{t \to 0} \frac{{\sin 3t}}{{\sin t}} \cr
& {\text{First rewrite the function by multiplying and dividing by }}3t \cr
& \frac{{\sin 3t}}{{\sin t}} = \frac{{\sin 3t}}{{\sin t}} \times \frac{{3t}}{{3t}} \cr
& \frac{{\sin 3t}}{{\sin t}} = \left( {\frac{{\sin 3t}}{{3t}}} \right)\left( {\frac{{3t}}{{\sin t}}} \right) \cr
& {\text{Therefore}}{\text{,}} \cr
& \mathop {\lim }\limits_{t \to 0} \frac{{\sin 3t}}{{\sin t}} = \mathop {\lim }\limits_{t \to 0} \left[ {\left( {\frac{{\sin 3t}}{{3t}}} \right)\left( {\frac{{3t}}{{\sin t}}} \right)} \right] \cr
& {\text{Use the product law of limits}}{\text{, the limit of a product is the }} \cr
& {\text{product of the limits}}{\text{, then}} \cr
& = \mathop {\lim }\limits_{t \to 0} \left( {\frac{{\sin 3t}}{{3t}}} \right)\mathop {\lim }\limits_{t \to 0} \left( {\frac{{3t}}{{\sin t}}} \right) \cr
& = 3\mathop {\lim }\limits_{t \to 0} \left( {\frac{{\sin 3t}}{{3t}}} \right)\mathop {\lim }\limits_{t \to 0} \left( {\frac{t}{{\sin t}}} \right) \cr
& = 3\mathop {\lim }\limits_{t \to 0} \left( {\frac{{\sin 3t}}{{3t}}} \right)\mathop {\lim }\limits_{t \to 0} {\left( {\frac{{\sin t}}{t}} \right)^{ - 1}} \cr
& {\text{Use the power law of limits }}\mathop {\lim }\limits_{x \to a} {\left[ {f\left( x \right)} \right]^n} = {\left[ {\mathop {\lim }\limits_{x \to a} f\left( x \right)} \right]^n} \cr
& = 3\mathop {\lim }\limits_{t \to 0} \left( {\frac{{\sin 3t}}{{3t}}} \right){\left[ {\mathop {\lim }\limits_{t \to 0} \left( {\frac{{\sin t}}{t}} \right)} \right]^{ - 1}} \cr
& {\text{Letting }}\theta = 3t,{\text{ then }}\theta \to 0{\text{ as }}x \to 0,{\text{ so by }}\mathop {\lim }\limits_{\theta \to 0} \frac{{\sin \theta }}{\theta } = 1 \cr
& = 3\overbrace {\mathop {\lim }\limits_{t \to 0} \left( {\frac{{\sin 3t}}{{3t}}} \right)}^{\theta = 3t,{\text{ }}\mathop {\lim }\limits_{\theta \to 0} \frac{{\sin \theta }}{\theta } = 1}\overbrace {{{\left[ {\mathop {\lim }\limits_{t \to 0} \left( {\frac{{\sin t}}{t}} \right)} \right]}^{ - 1}}}^{{1^{ - 1}}} \cr
& = 3\left( 1 \right)\left( 1 \right) \cr
& = 3 \cr
& {\text{Then}} \cr
& \mathop {\lim }\limits_{t \to 0} \frac{{\sin 3t}}{{\sin t}} = 3 \cr} $$
