Mathematics

Problem

Consider the behaviour of

\dfrac{\sin x}{x}

near

x=0

. a. (i) State the two standard trigonometric limits that involve

\sin x

and

\cos x

x

approaches

0

. b. (ii) Using a suitable geometrical or analytical argument, justify that

\displaystyle\lim_{x\to 0}\dfrac{\sin x}{x}=1

. Clearly mention each logical step involved. c. (iii) Hence or otherwise, determine

\displaystyle\lim_{x\to 0}\dfrac{\sin kx}{\sin mx}

, where

k

and

m

are non-zero real constants, and explain why your result is valid.