§ 1-4 Limits and Continuity

1. § 1-4 Limits and Continuity The student will learn about: limits, limits, finding limits, one-sided limits, infinite limits, and continuity.

2. Limits The word “limit” is used in everyday conversation to describe the ultimate behavior of something, as in the “limit of one’s endurance” or the “limit of one’s patience.” In mathematics, the word “limit” has a similar but more precise meaning.

3. Limits Given a function f(x), if x approaching 3 causes the function to take values approaching (or equalling) some particular number, such as 10, then we will call 10 the limit of the function and write In practice, the two simplest ways we can approach 3 are from the left or from the right.

4. Limits For example, the numbers 2.9, 2.99, 2.999, ... approach 3 from the left, which we denote by x→3 –, and the numbers 3.1, 3.01, 3.001, ... approach 3 from the right, denoted by x→3 +. Such limits are called one-sided limits.

5. Example 1 – FINDING A LIMIT BY TABLES Use tables to find Solution : We make two tables, as shown below, one with x approaching 3 from the left, and the other with x approaching 3 from the right.

6. x 2 2.9 2.99 2.999 3 3.001 3.01 3.1 4 f (x) 8 9.8 9.98 9.998 ? 10.002 10.02 10.2 12 Limits IMPORTANT! This table shows what f (x) is doing as x approaches 3. Or we have the limit of the function as x approaches We write this procedure with the following notation. 10 Def: We write 3 or as x → c, then f (x) → L if the functional value of f (x) is close to the single real number L whenever x is close to, but not equal to, c. (on either side of c). H

7. Limits As you have just seen the good news is that many limits can be evaluated by direct substitution.

8. Limit Properties These rules, which may be proved from the definition of limit, can be summarized as follows. For functions composed of addition, subtraction, multiplication, division, powers, root, limits may be evaluated by direct substitution, provided that the resulting expression is defined.

9. Examples – FINDING LIMITS BY DIRECT SUBSTITUTION Substitute 4 for x. Substitute 6 for x.

10. Example But be careful when a quotient is involved. But the limit exist!!!! Graph it. What happens at x = 2?

11. One-Sided Limit We have introduced the idea of one-sided limits. We write and call K the limit from the left (or left-hand limit) if f (x) is close to K whenever x is close to c, but to the left of c on the real number line. 5

12. One-Sided Limit We write and call L the limit from the right (or right-hand limit) if f (x) is close to L whenever x is close to c, but to the right of c on the real number line.

13. And a right-sided limit: The Limit Thus we have a left-sided limit: And in order for a limit to exist, the limit from the left and the limit from the right must exist and be equal.

14. Example f (x) = |x|/x at x = 0 0 The left and right limits are different, therefore there is no limit.

15. Infinite Limits Sometimes as x approaches c, f (x) approaches infinity or negative infinity. Consider From the graph to the right you can see that the limit is ∞. To say that a limit exist means that the limit is a real number, and since ∞ and - ∞ are not real numbers means that the limit does not exist.

16. Intro to Continuity As we have seen some graphs have holes in them, some have breaks and some have other irregularities. We wish to study each of these oddities. We will use our information of limits to decide if a function is continuous or has holes.

17. Continuity Intuitively, a function is said to be continuous if we can draw a graph of the function with one continuous line. I. e. without removing our pencil from the graph paper.

18. 1. f (c) is defined 2. 3. THIS IS THE DEFINITION OF CONTINUITY Definition A function f is continuous at a point x = c if

19. a. f (2) = 1 The limit exist! b. c. Example f (x) = x – 1 at x = 2. 1 2 Therefore the function is continuous at x = 2.

20. a. f (-3) = 0/0 Is undefined! - 6 b. c. Example f (x) = (x2 – 9)/(x + 3) at x = -3 -3 The limit exist! -6 Therefore the function is not continuous at x = -3. You can use table on your calculator to verify this.

21. Continuity Properties If two functions are continuous on the same interval, then their sum, difference, product, and quotient are continuous on the same interval except for values of x that make the denominator 0. Every polynomial function is continuous. Every rational function is continuous except where the denominator is zero.

22. Functions have three types of discontinuity. Consider - Graph on your calculator with a standard window. Continuity Summary. 1. Discontinuity at vertical asymptote. 2. Discontinuity at hole. 3. We have discontinuity with some functions that have a gap.

23. Summary. • We learned about limits and their properties. • We learned about left and right limits. • We learned about continuity and the properties of continuity.

24. ASSIGNMENT §1.4 On my website. 20, 21.