Trig Limits 6C9Ea9

1. **Problem Statement:** Find the limit of a function involving trigonometric expressions as the variable approaches a certain value. 2. **General Formula and Rules:** - Use the standard limit \(\lim_{x \to 0} \frac{\sin x}{x} = 1\). - For limits involving \(\cos x\), recall \(\lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2}\). - Apply algebraic manipulation and trigonometric identities to simplify the expression. 3. **Example:** Suppose we want to find \(\lim_{x \to 0} \frac{\sin 3x}{x}\). 4. **Step-by-step solution:** - Rewrite the limit as \(\lim_{x \to 0} \frac{\sin 3x}{3x} \cdot 3\). - Using the standard limit, \(\lim_{x \to 0} \frac{\sin 3x}{3x} = 1\). - Therefore, the limit is \(1 \times 3 = 3\). 5. **Explanation:** - We used the substitution \(u = 3x\) to transform the limit into a known standard form. - The factor outside the sine function scales the limit accordingly. 6. **Final answer:** \(3\).