Answer
$$\eqalign{
& {\text{The limit }}\mathop {\lim }\limits_{x \to 0} \frac{{\sin x}}{x}{\text{ was very useful in this section to calculate}} \cr
& {\text{trigonometric limits of the form }}\mathop {\lim }\limits_{x \to 0} \frac{{\sin ax}}{{\sin bx}},{\text{ }}\mathop {\lim }\limits_{x \to 0} \frac{{\sin ax}}{x},{\text{ and }} \cr
& {\text{it was used to prove the derivative of }}\sin x{\text{ using the definition}} \cr
& {\text{of the derivative}}{\text{.}} \cr} $$
Work Step by Step
$$\eqalign{
& {\text{The limit }}\mathop {\lim }\limits_{x \to 0} \frac{{\sin x}}{x}{\text{ was very useful in this section to calculate}} \cr
& {\text{trigonometric limits of the form }}\mathop {\lim }\limits_{x \to 0} \frac{{\sin ax}}{{\sin bx}},{\text{ }}\mathop {\lim }\limits_{x \to 0} \frac{{\sin ax}}{x},{\text{ and}} \cr
& {\text{it was used to prove the derivative of }}\sin x{\text{ using the definition}} \cr
& {\text{of the derivative}}{\text{.}} \cr} $$
