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1. Chapter 1 Limits of a Function Lesson 1: Limits of an Algebraic Function

2. Learning Outcomes Understand and apply the concept of limits in evaluating the behavior of a function as the independent variable approaches a specific value. Apply the limit theorems, including the limit of a constant, limit of a function, limit of sums, differences, products, and powers of functions, to evaluate limits and determine their existence.

3. Let’s Kick Off! Consider the given function: What is the (simplified) domain of ?  

4. Using a calculator, complete the table. 1.5 2.25 0.5   1.1 1.21 0.9   1.01 1.0201 0.99   1.001 1.002001 0.999   1.0001 1.00020001 0.9999   1.00001 1.0000200001 0.99999   1.5 2.25 0.5   1.1 1.21 0.9   1.01 1.0201 0.99   1.001 1.002001 0.999   1.0001 1.00020001 0.9999   1.00001 1.0000200001 0.99999  

5. Graph the given equation.

6. Definition: By the limit of a function f(x) as x approaches a, we shall mean the number L to which f(x) can be made as close as we wish by making x close enough to a. We denote the statement in the following manner:  

7. The main Limit Theorems In the definition of this theorem, let c be the constant, n as any positive integer and f and g as the given function, which has limit at a.

8. Limit of a Constant Theorem: Examples:   ,  

9. Limit of a function x: Examples:    

10. Limit of a constant and a function:

11. Limit of Sum and Difference of a Function:

12. Limit of Root of a Function:   Example:  

13. Limit of Product of Function:

14. Limit of Power of Function:

15. Limit of Quotient of a Function: Example:    

16. Lesson 2: Indeterminate Forms

17. F inding limits by conjugation

18. Lesson 3: Limit of Algebraic Function at Infinity

19. Limit at infinity are used to describe the behavior of functions as the independent variable increases or decreases without bound. When we say in calculus that something is “infinite”, we simply mean that there is no limit to its values.

20. Theorem. Limits at Infinity of Constant and Identity Function

21. Theorem. Limits at Infinity of a Constant Multiple of a Function

22. Example 1. Evaluate:    

23. Theorem. Infinite Limits Combinations ∞+∞ and ∞∙∞

24. Example 2: Evaluate:  

25. Example 3. Evaluate  

26. Example 4. Evaluate  

27. Example 5 Evaluate  

28. Example 6 Evaluate