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1. SOUTHERN LEYTE STATE UNIVERSITY

2. IC -Math 101 (Survey of Calculus)

3. MODULE 1: LIMITS OF FUNCTION

4. TOPIC 1: Limits of a n Algebraic Function

5. 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.

6. OVERVIEW Calculus is a branch of mathematics which started to develop in the 17 th Century. Sir Isaac Newton (English, 1642-1727) and Wilhelm Leibniz (German, 1646-1716) created calculus independently of each other and amazingly at about the same time. The word “calculus” is derived the Latin word for stone or pebble . Calculus roughly means a “method of calculus”.

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 Con stant and a Function :   Examples:    

11. Limit of Sum and Difference of a F unction:   Examples:        

12. Limit of Product of a Function:   Examples:      

14. Limit of Power of Function:   Examples:          

15. TOPIC 2: Indeterminate Forms

16. LEARNING OUTCOMES Apply different methods, including factoring and conjugation, to evaluate limits of rational functions and functions involving radicals. Understand and demonstrate the concept of indeterminate forms and how to handle them when evaluating limits.

17. Examples: 1. Evaluate     Solution: This cannot be evaluated by straight substitution since when x = 2, we have which is meaningless. That is, at x = 2 , the function assumes the indeterminate form . However, if x ≠ 2 , then  

18.   Therefore, to evaluate the limit of the given function, we proceed as follows:        

19. 2 . Evaluate if   Solution: A straight substitution of x = 2 leads to the indeterminate form . Since     Hence,    

21. 3.          

22. 4 .   Noticed that upon simplifying, the given function is undefined . It means that there is no existing limit . If we are going to observe, the denominator is radical. By conjugation method, we can cancel common factor and finally can apply the theorem of simplifying limits.  

24. TOPIC 3: Limit of Algebraic Function at Infinity

25. LEARNING OUTCOMES Evaluate limits involving "infinite" powers, roots, and denominators, using appropriate methods such as factoring and rationalizing. Understand and apply theorems related to limits at infinity, including limits of polynomial and rational functions, reciprocals of powers, and combinations of functions.

26. OVERVIEW Limit at infinity is 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.

27. Theorem : Limits at Infinity (for limits at Infinity) Supposed   1.   2.   3.   4.  

28. Theorem: Limits at Infinity of Constant and Identity Function Let c be any constant. Then 1 .   2 .  

29. Theorem: Limits at Infinity of a Constant Multiple of a Function Let n be a nonzero constant and . Then   1.   2.  

30. Theorem: Limits at Infinity of a Constant Multiple of a Function Example 1. Evaluate:   Solution: As , clearly (tautology). Now since , we thus have    

31. Theorem: Infinite Limits Combinations and   Supposed   1. If     2. If    

32. Theorem: The Infinite Limit Combination and   Supposed   1.   2 .   3 .  

33. Theorem: The Infinite Limit Combination and   Note that   Solution: Now, as as well. Thus, by the previous theorem,   Example 2. Evaluate:  

34. Meanwhile, Thus,    

35. Theorem: Limits of Power Function at Infinity Let n be a positive integer. If 1 . n is even, then   2 . n is odd, then  

36. Theorem: Limits of Powers of Functions Let n be a positive integer and . If   1. n is even, then   2. n is odd, then  

37. Theorem: Infinite Limits of Roots and Functions Let n be a positive integer. If 1. n is even and   2. n is odd and      

38. Theorem: Infinite Limits of Roots and Functions Example 3. Evaluate   Solution: As    

39. Theorem: Limits of Negative-Exponent/Index Power/Root Function at Infinity Let n be a positive integer. Then . And   1. n is even,   2. n is odd,  

40. Theorem: Limits of Polynomial and Rational Function at Infinity   Example 4. Evaluate   Solution: Factoring the highest power of x out of this expression, we have

41.   By our result on powers, Moreover, since all reciprocals of positive integer powers approach zero.   Finally then,  

42. Evaluate   Example 5   Solution: We factor out the highest powers up and down the fraction:  

43. Evaluate   Example 6   Solution: Again, we factor out the highest powers in the numerator and denominator:

44. Now, we just have to be careful about . This is not necessarily x ; it depends on the sign of x . Recall that, since the principal square root is the positive square root, we then have, in general     Now since , we then have . Thus,  

46. Theorem: Limits of Reciprocals of “Infinite” Powers/Roots Let n be a positive integer. Then .   1. n is even and or   2. n is even and   If either

47. Theorem: Limits Involving “Infinite” Denominators Supposed and Then    

48. Theorem: Limits Involving “Infinite” Denominators Evaluate   Example 7 Solution: As    

49. Thus, so that   Finally then,