A limit in introductory real analysis is a fundamental concept in mathematics that describes the behavior of a function as its input approaches a certain value. It is used to determine the value that a function approaches as its input gets closer and closer to a specific value, but may not necessarily reach that value.
To evaluate a limit in introductory real analysis, you must first determine the value that the function approaches as its input gets closer and closer to the given value. This can be done by plugging in values that are approaching the given value from both sides and seeing if the function approaches the same value, or by using limit laws and algebraic manipulation to simplify the expression.
There are several common techniques used to evaluate limits in introductory real analysis, including direct substitution, factoring, rationalization, and using L'Hospital's rule. These techniques are used to manipulate the expression and simplify it in order to determine the limit.
When evaluating limits in introductory real analysis, it is important to keep in mind the properties of limits, such as the limit laws and the Squeeze Theorem. It is also important to consider the behavior of the function as the input approaches the given value, and to be aware of any potential indeterminate forms that may require further simplification.
Limits are important in introductory real analysis because they allow us to understand the behavior of a function near a specific value, even if the function is not defined at that value. They are essential in determining continuity, differentiability, and other important properties of functions in mathematics and other fields of science.