1.4 Continuity & Limits

1. 1.4 Continuity & Limits Ms. Hernandez Calculus

2. Calculus • 1.4 Continuity and 1-sided limits • Student objectives: • Understand & describe & find continuity at a point vs continuity on an open interval • Find 1-sided limits • Use properties of continuity • Understand & use the Intermediate Value Theorem

3. Continuity • Continuous line • Trace ur finger along the line • If there are no breaks along the line • No jumping allowed! • No breaking allowed! • No gaps allowed! • Line = function = f(x)

4. Continuity example • P68, see fig 1.25 – 3 graphs • Discuss w/group • Why are the 3 graphs not continuous at x = c? Graph 1 – gap in the graph function is not defined at x = c Graph 2 – break in the graph the lim of f(x) dne at x = c Graph 3 – jump in the graph the lim of f(x) exists at x=c but does not equal f(c)

5. Def of Continuity Continuous at a point if: 1. function is defined at x = c 2. the lim of f(x) at x = c 3. the lim of f(x) exists at x=c and equals f(c) p68

6. Def of Continuity • Continuity on an open interval • (a,b) some points on a graph or f(x) • Interval includes everything in between a & b • So that’s why we say on an open interval • But what is that?

7. Def of Continuity • If you can trace your finger all along the line on the interval (a,b) then its conts. • Include a & b. • So no gaps, jumps, or breaks. • If there is a gap/jump/break then f(x) has a discontinuity • Removable discontinuity (can b fixed easily) • P69 Fig 1.26(a),(c) • Non-removable discontinuity (not 2 easy) • P69 Fig 1.26 (b)

8. Continuity of a function • Discuss example 1 pg 69 • How does the domain clue you into f(x) continuity? Not defined then its not conts there! Vertical asymptotes where domain is undefined for f(x)

9. 1-sided limits • Limit from the left (-) • Limit from the right (+) • When working with square roots take from the + • p70

10. Continuity on a closed interval • A function is continuous on the closed interval if • Lim from left exists • Lim from right exists • Ex 4 pg 71

11. Charles’s Law & Absolute Zero • Read over example 5 on pg 72 • What steps would you take to find absolute zero on the Fahrenheit scale • What property must the temp scale have, according to Charles’s Law

12. Thm 1.11 Properties of Continuity • p73 • Scalar • Sum & difference • Product • Quotient • All polynomial (in their domain) • All rational (in their domain) • All radical (in their domain) • All trig (in their domain) • So how many functions are conts at every point in their domain?

13. Consequences of Continuity • Composite functions • Thm 1.12 if 2 functions f, g are conts then their composites are also conts • What’s the consequence of the limit of this composite function? • Is this any different from last lecture? • If so, how?

14. Testing for continuity • Describe intervals that f(x) is continuous on • Squeeze thm & continuity

15. IVT: Intermediate Value Theorem • Tells you why/what but not HOW • Existence theorem • If f is conts on the closed interval [a,b] and k is any number between f(a) and f(b) (ON THE GRAPH so y = k somewhere between endpoints a & b) then there is at least one number c (that we can input into function) in [a,b] such that f(c) = k.

16. APPLY IVT • Use the IVT to show that the polynomial function f(x) = x2 + 2x –1 has a zero in the interval [0,1]. • We know f(x) is continuous on [0,1] b/c • f(0) = 02 + 2(0) –1 = -1 -1 is on the graph • Testing left endpoint • f(1) = 12 + 2(1) –1 = 2 2 is on the graph • Testing right endpoint • f(0) < 0  -1 < 0 • f(1) > 0  2 > 0 • So zero is on the graph in between the endpoints [a,b] • So then some number c could be plugged into f(x) and get 0 since 0 in between f(a) and f(b) or [a,b]. • See Ex 8 on pg 75

17. Some examples 1.4 #3-6. I can tell from the graph that f(x) is not conts at x = c. If I ask you on a non-calculator quiz/test to verify continuity of f(x) how would you do that? We must examine the 3 parts of the definition of continuity. So which part of the continuity theorem is f(x) failing? • #3, #4 • Is f(c) defined? Yes f(c) is defined. • Does the limit exist at f(c)? Yes, it does! The left and right limits are equal. • Does the limit value at c = f(c)? Nope! • #5,#6 • Is f(c) defined? Yes f(c) is defined. • Does the limit exist at f(c)? Nope, left and right limits are not equal. • STOP.

18. 1.4 #9 • lim x  -3- = ? • Vertical asymptotes always occur at points where the domain is not defined • http://www.purplemath.com/modules/asymtote.htm • lim x  -3- = DNE b/c grows without bound • Does your graphing calculator help?

19. 1.4 #41 • Domain does not include –2 or 5 • Lim x  5 DNE • Non-removable discontinuity • Indeterminate form • Lim x -2 = -1/7 • Removable discontinuity • Even though –2 is not in the domain, (x+2) can be factored out and you get an answer of –1/7 for the limit • Does your calculator help you?

20. 1.4 # 57) Find constants a & b such the the function is conts on the entire real line f(2) = 8 So I need to find a such that Because then my piecewise function would be continuous. In solving for a, I set it equal to the limit value I need. When I calculate a limit I would like to direct sub, so I plug in 2. Solving, a must be 2.

21. 1.4 #75 Explain why the function has a zero in the specified interval • f(x) is continuous on what interval? • Evaluate the endpoints of the interval? • Is zero on the graph in between the endpoints? is continuous on the interval [1,2] f(1) = 33/16 f(2) = -4 So f(c) = 0 works and zero is in the interval for at least one value of c between 1 and 2.

22. 1.4 #100-103 • 100 – True all conditions are met f(c) = L is defined, lim xc = L exists, and f(c) = lim xc • 101 - True b/c one of the limits does not equal the function at x = c • 102 – False b/c a rational function can be written as P(x)/Q(x) where P and Q are polynomials of degree m and n, respectively. So it will at most n discontinuities. • 103 - False b/c f(1) is not defined and lim x1 DNE