\lim_{x \to 0} \frac{\sin(3x)}{x} = 3 \lim_{x \to 0} \frac{\sin(3x)}{3x} = 3 \times 1 = 3

["Understanding the Limit: (\lim_{x \ o 0} \frac{\sin(3x)}{x} = 3)", "When encountering limits in calculus, one of the most fundamental and powerful identities to remember is the basic limit:", "[\n\lim_{u \ o 0} \frac{\sin u}{u} = 1\n]", "This foundational result allows us to simplify and evaluate complex trigonometric limits by manipulating expressions into familiar forms. One important form arises when limits involve arguments other than just (x), such as (\lim_{x \ o 0} \frac{\sin(3x)}{x}).", "Evaluating (\lim_{x \ o 0} \frac{\sin(3x)}{x})", "At first glance, plugging (x = 0) into (\frac{\sin(3x)}{x}) yields the indeterminate form (\frac{0}{0}), requiring deeper analysis. To resolve this, we rewrite the expression to make use of the well-known limit.", "We begin by separating the constant factor inside the sine function:", "[\n\frac{\sin(3x)}{x} = \frac{\sin(3x)}{3x} \cdot 3\n]", "Now take the limit as (x \ o 0):", "[\n\lim_{x \ o 0} \frac{\sin(3x)}{x} = \lim_{x \ o 0} \left( \frac{\sin(3x)}{3x} \cdot 3 \right)\n]", "According to the limit properties, we can factor the constant (3):", "[\n= 3 \cdot \lim_{x \ o 0} \frac{\sin(3x)}{3x}\n]", "Now observe that the expression (\frac{\sin(3x)}{3x}) approaches the fundamental limit as its argument approaches zero. Substitute (u = 3x). As (x \ o 0), (u \ o 0) too, and:", "[\n\lim_{x \ o 0} \frac{\sin(3x)}{3x} = \lim_{u \ o 0} \frac{\sin u}{u} = 1\n]", "Therefore:", "[\n\lim_{x \ o 0} \frac{\sin(3x)}{x} = 3 \cdot 1 = 3\n]", "Why This Identity Matters", "This identity—(\frac{\sin(kx)}{x} = k \cdot \frac{\sin(kx)}{kx})—extends the useful limit (\lim_{u \ o 0} \frac{\sin u}{u} = 1) to any linear argument. It simplifies many trigonometric limit problems in calculus and is essential for deriving derivative formulas involving sine and cosine.", "Conclusion", "The limit (\lim_{x \ o 0} \frac{\sin(3x)}{x} = 3) exemplifies how algebraic manipulation and a deep understanding of basic trigonometric limits enable elegant solutions. By recognizing patterns and applying limit properties, complex expressions become manageable—and this limit is a key building block in advanced calculus.", "Keywords:\n(\lim_{x \ o 0} \frac{\sin(3x)}{x}), trigonometric limits, (\lim_{u \ o 0} \frac{\sin u}{u} = 1), calculus, limit evaluation, math, trigonometry, fundamental limit, derivative calculation."]