Computing Limits — Question 9

Question 9 - Solution

We evaluate $$\lim_{x \to 0} \frac{\sin(3x)}{\tan(5x)}.$$

Step 1: Express tangent in terms of sine and cosine

Recall that $$\tan(5x) = \frac{\sin(5x)}{\cos(5x)}.$$ Thus, $$\frac{\sin(3x)}{\tan(5x)} = \frac{\sin(3x)\cos(5x)}{\sin(5x)}.$$

Step 2: Apply standard trigonometric limits

Rewrite the expression as $$\frac{\sin(3x)}{3x} \cdot \frac{5x}{\sin(5x)} \cdot \frac{3}{5} \cdot \cos(5x).$$

Now take limits term by term as $x \to 0$: $$\frac{\sin(3x)}{3x} \to 1, \quad \frac{5x}{\sin(5x)} \to 1, \quad \cos(5x) \to 1.$$

Therefore, $$\lim_{x \to 0} \frac{\sin(3x)}{\tan(5x)} = 1 \cdot 1 \cdot \frac{3}{5} \cdot 1 = \frac{3}{5}.$$

$$\boxed{\frac{3}{5}}$$