91Ó°ÊÓ

The concept of the sum of limits is quite straightforward and is one of the fundamental properties of limits in calculus. When dealing with the limit of a sum of two functions, the rule is that you can simply add their individual limits together, provided these limits exist. This is known as the sum rule for limits. If you have two functions, say \( f(x) \) and \( g(x) \), and you know their limits as \( x \) approaches a certain value exist, then the limit of their sum can be determined by adding their limits.

As demonstrated in the exercise:

• We had \( \lim _{x \rightarrow 3} f(x) = 7 \)
• and \( \lim _{x \rightarrow 3} g(x) = 5 \).

Thus, \( \lim _{x \rightarrow 3} (f(x) + g(x)) = 7 + 5 = 12 \). This property helps simplify complex limit problems by breaking them into manageable parts.

Understanding limit properties is crucial for efficiently solving calculus problems. Certain properties can make calculating limits much easier:

• Linearity: The limit of a sum is the sum of the limits. This is our sum rule, which simplifies limit evaluation by allowing the separation of sums into individual limits.
• Scaling Constant Function: If \( k \) is a constant, then \( \lim_{x \to a} k \cdot f(x) \) is \( k \cdot \lim_{x \to a} f(x) \).
• Continuous Functions: The limit of a continuous function at a point is just the function's value at that point.

These properties allow you to work through limits and simplify complex expressions. Notably, they help in validations like in our original problem, where the knowledge that the sum of limits directly gives the limit of a sum allows us to confidently evaluate the problem step by step.

Limit evaluation involves using the properties and rules of limits to find a limit's value. You begin by identifying the type of limit you're facing, whether it's a direct substitution, a need to factor, or an application of rules like L'Hôpital's Rule.

In problems like the given exercise:

• We evaluated the limit using the sum rule. The task was simplified to just adding known values \( \lim _{x \to 3} f(x) \) and \( \lim _{x \to 3} g(x) \).
• No complex methods were necessary because these limits were already directly given. This is a prime example where basic properties suffice for evaluation.

The key is to use methods that relate directly to the given information, ensuring you apply simple calculations whenever possible. When limits are known or can be derived straightforwardly, evaluation can remain uncomplicated, as seen here.