Evaluate limit: \( \lim_{x o 2} 2(x + 2) = 2(4) = 8 \).

["# Evaluate the Limit: ( \lim_{x \ o 2} 2(x + 2) = 8 )", "Understanding how to evaluate limits is a foundational skill in calculus, and simplifying expressions plays a key role in this process. One commonly encountered limit involves evaluating a continuous function at a specific point by direct substitution. In this article, we’ll carefully analyze the limit:", "[\n\lim_{x \ o 2} 2(x + 2)\n]", "and verify that it correctly evaluates to 8 using algebraic evaluation and the Limit Laws.", "## What Does ( \lim_{x \ o 2} 2(x + 2) ) Mean?", "The expression ( \lim_{x \ o 2} 2(x + 2) ) asks: what value does the function ( f(x) = 2(x + 2) ) approach as ( x ) gets arbitrarily close to 2 from both sides? Since ( f(x) ) is a continuous function everywhere on the real line, we can directly substitute ( x = 2 ) into the expression.", "## Evaluating the Limit by Substitution", "Start with the function:", "[\nf(x) = 2(x + 2)\n]", "Now, apply direct substitution by replacing ( x ) with 2:", "[\nf(2) = 2(2 + 2)\n]", "Simplify inside the parentheses:", "[\n= 2(4) = 8\n]", "Thus,", "[\n\lim_{x \ o 2} 2(x + 2) = 8\n]", "## Applying the Limit Laws for Confirmation", "While direct substitution works here due to continuity of ( f(x) = 2(x + 2) ), we can also confirm this result rigorously using the Limit Laws from calculus.", "The Limit Law of Constants states that multiplying a constant by the limit of a function equals the constant multiplied by that limit:", "[\n\lim_{x \ o a} c \cdot g(x) = c \cdot \lim_{x \ o a} g(x)\n]", "Apply this to ( f(x) = 2 \cdot (x + 2) ):", "[\n\lim_{x \ o 2} 2(x + 2) = 2 \cdot \lim_{x \ o 2} (x + 2)\n]", "Now apply the Limit Law for addition:", "[\n\lim_{x \ o 2} (x + 2) = \lim_{x \ o 2} x + \lim_{x \ o 2} 2 = 2 + 2 = 4\n]", "Now multiply by the constant:", "[\n2 \cdot 4 = 8\n]", "So the limit confirms:", "[\n\lim_{x \ o 2} 2(x + 2) = 8\n]", "## Why Evaluating Limits Like This Matters", "Evaluating limits at specific points using direct substitution is essential in calculus because it ensures clarity and efficiency in solving problems involving continuity, derivatives, and integrals. Recognizing when direct substitution is valid (such as with continuous functions like polynomials and linear expressions) saves time and reduces complexity.", "## Summary", "The limit", "[\n\lim_{x \ o 2} 2(x + 2)\n]", "is evaluated effectively by direct substitution or step-by-step application of the Limit Laws, yielding:", "[\n\boxed{8}\n]", "This illustrates a simple but important concept in calculus: continuous functions have predictable, smooth behavior near their defined points, and limits help quantify this behavior precisely. Whether studying for exams or solving practical problems, mastering limit evaluation builds a strong foundation in mathematical analysis.", "---", "Keywords: limit evaluation, calculate limit, ( \lim_{x \ o 2} 2(x + 2) ), direct substitution, limit laws, calculus basics, continuous functions, algebra and limits.\nMeta Description: Learn how to evaluate ( \lim_{x \ o 2} 2(x + 2) ) step-by-step using direct substitution and Limit Laws, with clear explanation and verification."]