Home » Blog » Find the limit as x goes to 0 of (sin (ax)) / sin (bx))

Find the limit as x goes to 0 of (sin (ax)) / sin (bx))

by RoRi

Evaluate the limit.

    \[ \lim_{x \to 0} \frac{\sin (ax)}{\sin (bx)}. \]

We know (p. 287) that the expansion for \sin x is given by

    \[ \sin x = x + o(x^2) \qquad \text{as} \qquad x \to 0. \]

Therefore,

    \begin{align*} \lim_{x \to 0} \frac{\sin (ax)}{\sin (bx)} &= \lim_{x \to 0} \frac{ax + o(x^2)}{bx + o(x^2)} \\[9pt] &= \lim_{x \to 0} \left( \frac{a}{b} \cdot \frac{bx + o(x^2)}{bx+o(x^2)} \right) \\[9pt] &= \frac{a}{b}. \end{align*}

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):