Evaluate the following limits

MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

Evaluate the following limits, showing each step clearly. (i)

\displaystyle\lim_{x\to 0}\dfrac{\sin 7x}{7x}

(ii) Using the answer to part (i), find

\displaystyle\lim_{x\to 0}\dfrac{\sin7x}{x}

. (iii) Hence calculate

\displaystyle\lim_{x\to 0}\dfrac{\sin 7x}{\sin 5x}

. Interpret your result briefly.