Evaluate the limit: \( \lim_{x o 3} (x + 3) = 3 + 3 = 6 \).
["Evaluate the Limit: Understanding Why ( \lim_{x \ o 3} (x + 3) = 6 )", "When working with limits in calculus, one of the most straightforward evaluations involves direct substitution—especially when dealing with continuous functions. Consider the limit:", "[\n\lim_{x \ o 3} (x + 3)\n]", "This limit asks: What value does the expression ( x + 3 ) approach as ( x ) gets arbitrarily close to 3 from both the left and the right?", "### Applying Direct Substitution", "Since the function ( f(x) = x + 3 ) is a simple linear (and therefore continuous) function over the real numbers, we can apply the fundamental rule of limits:", "[\n\lim_{x \ o a} f(x) = f(a)\n]", "Here, ( a = 3 ). Applying this rule:", "[\n\lim_{x \ o 3} (x + 3) = 3 + 3 = 6\n]", "There’s no need for complex trigonometric limits, L’Hôpital’s Rule, or方向一般，此处继续规范表达：\n因为该函数为线性函数且在 ( x = 3 ) 处连续，因此可以直接将 ( x ) 替换为 3，计算得出该极限为 6。", "### Why Direct Substitution Works", "The function ( f(x) = x + 3 ) has no discontinuities, holes, or vertical asymptotes at ( x = 3 ). As ( x ) approaches 3, the expression ( x + 3 ) smoothly approaches ( 6 ). This aligns with the definition of a limit at a point for continuous functions.", "### Conclusion", "Therefore, the evaluation:", "[\n\lim_{x \ o 3} (x + 3) = 3 + 3 = 6\n]", "is correct and reflects a foundational principle in calculus—continuous functions preserve limits under direct substitution. This simple yet powerful concept forms the backbone for evaluating more complex limits and understanding function behavior near specific points.", "Keywords: ( \lim_{x \ o 3} (x + 3) ), limit evaluation, continuous functions, calculus basics, direct substitution, math explanation.", "---", "This article provides a clear, beginner-friendly explanation of evaluating limits with continuous functions, ideal for students learning foundational calculus concepts."]
