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Limit evaluation is a fundamental concept in calculus where we determine the value that a function approaches as the input gets close to a certain point. For example, in the exercise, we are evaluating limits as \(x\) approaches 2. The limit helps us understand the behavior of a function near a specific input value, rather than just at the point itself.
In the given exercise, each expression—\(3 + x\), \(3 + x\) without parentheses, and \(x + 3\)—should be evaluated as \(x\) approaches 2. By substituting \(x = 2\), we find that all these functions equate to 5.
This substitution method works because the function is continuous at \(x = 2\), meaning there's no sudden break or jump. An essential insight here is:

• When evaluating limits of continuous functions, direct substitution is often enough.
• Whether the expression is \(3 + x\) or \(x + 3\), direct substitution of \(x\) gives the limit.
• Understanding that the expressions are essentially the same helps simplify why they yield the same result when substituting \(x = 2\).

The commutative property is a basic principle in algebra which states that the order of numbers in addition or multiplication does not affect the result. In mathematical terms, for addition, this is expressed as \(a + b = b + a\). This means that no matter how we group the numbers when adding them, the sum remains the same.
In the context of limits, when examining expressions such as \(3 + x\) and \(x + 3\), the commutative property assures us that they are essentially identical. Therefore, evaluating limits for these kinds of expressions will yield identical results.
This property can greatly simplify limit problems:

• Recognizing commutativity can help identify when expressions are equivalent, easing computations.
• It provides a foundation for flexible manipulation of mathematical expressions without altering their intrinsic value.

Function substitution is a straightforward technique used in calculus to evaluate limits. It involves directly replacing the variable in a function with a given value. This is particularly useful when dealing with continuous functions, where direct evaluation at a point reflects the behavior of the function around that point.
In the exercise, substituting \(x = 2\) into each expression—whether \(3 + x\), \(3 + x\) without grouping, or \(x + 3\)—shows that they all resolve to 5. This illustrates that function substitution provides a convenient method for determining limits in simple cases.
A few key takeaways when using function substitution are:

• Ensure the function is continuous at the point of substitution, as discontinuities might skew results.
• Recognize that substitution is an effective first step in analyzing the behavior of a function at a specific limit.
• In this exercise, recognizing the expressions result in the same value through substitution simplifies the evaluation of the limit.