AP Calculus AB and BC

Limit of sin(4x)/sin(2x) as x Approaches 0

Q: Does this work for tangents too?

Yes. Since \frac{\tan u}{u} \to 1 as well, quotients mixing sine and tangent with linear inner angles reduce to the same ratio of coefficients.

Q: What if the inner angle is not linear?

The shortcut fails. For something like \frac{\sin(x^{2})}{\sin x} you have to match each piece to its own inner expression, and the answer is 0 rather than a ratio.

The limit of sine of 4x over sine of 2x as x approaches zero is two. Convert the top and the bottom separately to the standard sine limit, and the leftover constants four and two divide to give the answer.

\lim_{x \to 0} \frac{\sin 4x}{\sin 2x} = 2

Settled by matching each inner angle to the standard limits.

Convert both parts

Divide top and bottom by $x$ , then arrange each piece into the standard shape $\frac{\sin u}{u}$ .

\frac{\sin 4x}{\sin 2x} = \frac{\dfrac{\sin 4x}{x}}{\dfrac{\sin 2x}{x}} = \frac{4 \cdot \dfrac{\sin 4x}{4x}}{2 \cdot \dfrac{\sin 2x}{2x}}

Each fraction inside tends to 1, leaving $\frac{4}{2} = 2$ .

The shortcut, and when it is safe

For a quotient of sines with linear inner angles, the answer is just the ratio of the coefficients: $\frac{\sin ax}{\sin bx} \to \frac{a}{b}$ . That covers a large share of exam questions in one step.

It is only valid at $x \to 0$ , where both sines vanish. Away from 0 there is no indeterminate form and direct substitution is the correct move.

Not sure which technique a limit wants?

The Limit Method Chooser walks the decision from direct substitution through factoring, the conjugate, and L'Hopital, and says why each one applies or fails.

Frequently asked questions

Does this work for tangents too?

Yes. Since $\frac{\tan u}{u} \to 1$ as well, quotients mixing sine and tangent with linear inner angles reduce to the same ratio of coefficients.

What if the inner angle is not linear?

The shortcut fails. For something like $\frac{\sin(x^{2})}{\sin x}$ you have to match each piece to its own inner expression, and the answer is 0 rather than a ratio.