Evaluate the limit \(\lim_{x \to 3} \frac{x^2 - 9}{x - 3}\).
["Evaluate the Limit: (\lim_{x \ o 3} \frac{x^2 - 9}{x - 3})", "Understanding limits is fundamental in calculus, and evaluating one common limit provides clear insight into function behavior near a point. In this article, we will evaluate:", "[\n\lim_{x \ o 3} \frac{x^2 - 9}{x - 3}\n]", "---", "### Step 1: Recognize the Indeterminate Form", "At first glance, substituting (x = 3) into the expression gives:", "[\n\frac{3^2 - 9}{3 - 3} = \frac{0}{0}\n]", "This is an indeterminate form, meaning direct substitution does not yield a result. To evaluate this limit, we must simplify the expression before applying limits.", "---", "### Step 2: Factor the Numerator", "The numerator (x^2 - 9) is a difference of squares, which factors as:", "[\nx^2 - 9 = (x - 3)(x + 3)\n]", "Substitute this factorization into the original limit:", "[\n\lim_{x \ o 3} \frac{(x - 3)(x + 3)}{x - 3}\n]", "---", "### Step 3: Simplify the Expression", "For all (x <br/>\neq 3), the factor (x - 3) in the numerator and denominator cancels out:", "[\n\lim_{x \ o 3} (x + 3)\n]", "This simplification is valid near (x = 3), excluding (x = 3) itself, but since limits consider values approaching a point, removing the discontinuity simplifies evaluation.", "---", "### Step 4: Evaluate the Simplified Limit", "Now compute the limit of the simplified expression:", "[\n\lim_{x \ o 3} (x + 3) = 3 + 3 = 6\n]", "---", "### Final Answer", "[\n\boxed{6}\n]", "---", "### Why This Limiting Behavior Matters", "Although the function (\frac{x^2 - 9}{x - 3}) is undefined at (x = 3), its limit exists and equals 6. This demonstrates how a removable discontinuity (a "hole" in the graph) allows the function to approach a definite value even where it is not defined. Recognizing and simplifying such limits is essential in calculus, particularly when analyzing continuity, derivatives, and integrals.", "---", "### SEO Optimization", "- Keywords: limit evaluation, (\lim_{x \ o 3} \frac{x^2 - 9}{x - 3}), indeterminate form, factoring, simplifying limits, calculus fundamentals\n- Semantic Relevance: Clear explanation, step-by-step solving, conceptual insight\n- User Intent: Students and learners seeking to understand limit evaluation using algebraic techniques and real-world calculus concepts.", "---", "By mastering this type of limit, you build a strong foundation for more advanced calculus topics while improving problem-solving precision. Always simplify algebraically before applying limit laws—this approach ensures accuracy and clarity."]
