The Limit — Question 10

Question 10

Evaluate the limit: $$\lim_{x \to 0} \frac{\sin(3x)}{x}$$

Question 10 - Solution

We are given: $$\lim_{x \to 0} \frac{\sin(3x)}{x}$$

We want to simplify the expression to match a known limit. Recall the standard limit: $$\lim_{u \to 0} \frac{\sin(u)}{u} = 1$$

Let $u = 3x$. As $x \to 0$, we also have $u \to 0$.

So: $$\frac{\sin(3x)}{x} = \frac{\sin(3x)}{3x} \cdot 3$$

Now apply the limit: $$\lim_{x \to 0} \frac{\sin(3x)}{x} = \lim_{x \to 0} \left( \frac{\sin(3x)}{3x} \cdot 3 \right) = \left( \lim_{u \to 0} \frac{\sin(u)}{u} \right) \cdot 3 = 1 \cdot 3 = \boxed{3}$$