\(\lim_{x \to 0} \frac{\sin(3x)}{x}\) を求めなさい。
["How to Evaluate (\lim_{x \ o 0} \frac{\sin(3x)}{x}) and Why It Matters", "Understanding limits is fundamental in calculus, and one classic example is (\lim_{x \ o 0} \frac{\sin(3x)}{x}). This limit not only tests key concepts in trigonometry and calculus but also reveals important properties of trigonometric functions near zero. In this SEO-optimized article, we’ll explore how to find this limit step-by-step, why it equals 3, and its relevance in mathematical applications.", "---", "### Understanding the Limit", "The limit in question is:", "[\n\lim_{x \ o 0} \frac{\sin(3x)}{x}\n]", "As (x) approaches 0, both the numerator (\sin(3x)) and the denominator (x) approach 0. This creates an indeterminate form (\frac{0}{0}), which means we cannot directly substitute (x = 0). Instead, we rely on limit properties and known trigonometric limits to evaluate it.", "---", "### Applying Key Trigonometric Limits", "A standard result in calculus is:", "[\n\lim_{u \ o 0} \frac{\sin u}{u} = 1\n]", "To apply this, we manipulate the given expression to match this form. Start by rewriting the argument of the sine function:", "Let (u = 3x). As (x \ o 0), (u \ o 0) too. Substitute (x = \frac{u}{3}):", "[\n\lim_{x \ o 0} \frac{\sin(3x)}{x} = \lim_{u \ o 0} \frac{\sin(u)}{u/3} = \lim_{u \ o 0} 3 \cdot \frac{\sin u}{u}\n]", "Now, using the standard limit:", "[\n\lim_{u \ o 0} \frac{\sin u}{u} = 1 \implies \lim_{u \ o 0} 3 \cdot \frac{\sin u}{u} = 3 \cdot 1 = 3\n]", "---", "### Conclusion", "Thus,", "[\n\lim_{x \ o 0} \frac{\sin(3x)}{x} = 3\n]", "---", "### Why This Limit Is Important", "- Fundamental Build Block: This example demonstrates substitution and transformation techniques elementary in limit evaluation.\n- Connection to Derivative of Sine: It reflects the geometric definition used to derive (\lim_{u \ o 0} \frac{\sin u}{u} = 1), reinforcing the link between calculus and trigonometry.\n- Practical Applications: Such limits appear in physics and engineering when analyzing oscillations, waves, and small-angle approximations.", "---", "### Key SEO Keywords", "- (\lim_{x \ o 0} \frac{\sin(3x)}{x})\n- Evaluate limit\n- Trigonometric limit\n- Calculus limit tutorial\n- (\lim_{x \ o 0} \frac{\sin(kx)}{x})\n- Small angle limit\n- Calculus fundamentals", "---", "### Final Note", "Mastering limits like (\lim_{x \ o 0} \frac{\sin(3x)}{x}) strengthens your analytical skills and prepares you for advanced calculus, derivatives, and integration. Use this example as a foundation for exploring trigonometric identities and infinite limits.", "---", "Keywords: (\lim_{x \ o 0} \frac{\sin(3x)}{x} = 3), trigonometric limits, calculus tutorial, small angle limit, limit evaluation, standard limit (\frac{\sin u}{u} = 1)"]
