Section 2.7: Precise Definition of Limits

Limit of a Function

The concept of a limit is fundamental in calculus, providing a rigorous foundation for understanding how functions behave as their inputs approach specific values. The limit of a function as x approaches a is denoted as:

This means that for any number , there exists a corresponding number such that:

whenever

• Key Point: The definition ensures that the function values can be made arbitrarily close to the limit L by choosing x sufficiently close to a, but not equal to a.

• Application: This definition is used to rigorously prove statements about limits and to analyze the behavior of functions near specific points.

Steps for Proving a Limit

To prove that using the precise definition, follow these steps:

1. Find : Let be an arbitrary positive number. Use the inequality to find a condition of the form , where depends only on the value of .

2. Write a Proof: For any , assume and use the relationship between and found in Step 1 to prove that .

• Key Point: The proof must show that for every , a suitable can be found, ensuring the function values are within of the limit.

Example: Finding for a Linear Function

Suppose we want to find the largest value of such that for a given function and limit. Consider the function as approaches 1, with and .

• Set up the inequalities for near the limit:

• →

• →

• Therefore,

• Key Point: By solving for values that keep within of , we determine how close must be to (here, ) to satisfy the definition.

• Example Application: This process is used for more complex functions, adjusting the algebra to solve for in terms of .

Summary Table: Epsilon-Delta Definition Steps

The following table summarizes the steps for proving a limit using the epsilon-delta definition:

Additional info: The notes provided are directly relevant to Chapter 2: Limits, specifically focusing on the precise (epsilon-delta) definition of limits and its application in proofs and examples.