AP Calculus AB and BC

Limit of tan 3x / x as x Approaches 0 Is 3

Q: What is the general rule?

\lim_{x \to 0}\frac{\tan(ax)}{x} = a, the same coefficient the sine version produces.

Q: Why does the cosine not change the answer?

Because \cos 3x is continuous at 0 with value 1, so that factor contributes exactly 1.

The limit of tan 3x over x as x approaches 0 is 3. Split the tangent into sine over cosine: the sine part gives 3 by the special trigonometric limit and the cosine part goes to 1, so the product is 3.

\lim_{x \to 0} \frac{\tan 3x}{x} = 3

Settled by splitting tangent into sine over cosine.

Splitting the tangent

Tangent is sine over cosine, and separating them turns one unfamiliar limit into two familiar ones.

\frac{\tan 3x}{x} = \frac{\sin 3x}{x}\cdot\frac{1}{\cos 3x} = 3\cdot\frac{\sin 3x}{3x}\cdot\frac{1}{\cos 3x}

The middle factor tends to $1$ by the special limit, and cosine is continuous with $\cos 0 = 1$ , so the last factor tends to $1$ as well.

\lim_{x \to 0}\frac{\tan 3x}{x} = 3 \cdot 1 \cdot 1 = 3

The general shape

The limit of tan(ax) over x as x approaches 0 is a. It matches the sine version, because near the origin tangent and sine agree to first order.

Why the cosine factor is harmless

Splitting a limit into a product needs both pieces to converge, and here they do. The cosine factor is the easy one: it is continuous at $0$ with value $1$ , so it can simply be evaluated.

That is not true further out. At $x = \frac{\pi}{6}$ the inner angle reaches $\frac{\pi}{2}$ , where tangent is undefined, so this limit argument is strictly local to the origin.

The mistakes students make

Answering $1$ by pattern-matching the special limit without checking that the inner angle matches the denominator.
Answering $\frac{1}{3}$ by inverting the coefficient. The factor inside the tangent goes on top.
Assuming the limit is the same everywhere. Tangent has vertical asymptotes, so this only describes the behaviour near $0$ .

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

What is the limit of tan 3x / x as x approaches 0?

It is $3$ .

What is the general rule?

$\lim_{x \to 0}\frac{\tan(ax)}{x} = a$ , the same coefficient the sine version produces.

Why does the cosine not change the answer?

Because $\cos 3x$ is continuous at $0$ with value $1$ , so that factor contributes exactly $1$ .