MAT01A1: Precise Definition of a Limit and Continuity

Transcription

1 MAT01A1: Precise Definition of a Limit and Continuity Dr Craig 7 March 2018

2 Semester Test 1 D1 LAB 110 Be seated by 08h15. Everything up to and including Ch 2.3. Bring student cards. No bags. No calculators.

3 Test preparation Prescribed tutorial problems Class tests and assignments Saturday class worksheets Feedback from e-quizzes Khan Academy worksheets 2017 test paper Get help at the Tutor Centre Get help from a lecturer

4 Lecturers Consultation Hours Between now and the test: Thursday: 11h20 12h55 Dr Craig (C-508) Friday: 11h20 13h45 Dr Craig (C-508)

5 Today s lecture Precise definition of a limit Continuity

6 The precise definition of a limit (Chapter 2.4)

7 Definition of a limit of a function Definition: Suppose f(x) is defined for x near* a. If we can get the values of f(x) as close to L as we like by taking x as close to a on either side (but not equal to a), then we write lim f(x) = L x a and say the limit of f(x), as x approaches a, equals L *near: f(x) is defined on some open interval including a, but possibly not at a itself.

8 Before, we give the formal definition, let us think of the previous definition in a more informal way. Essentially, the definition of a limit is saying: If you give me any y value close to L, I can give you an x value such that f(x) will be closer than your y value to L.

9 The importance of absolute values In our first definition we said as close to L as we like and on the last slide we said a y value close to L Q: Do we want to allow the y value to be a little bit below L or a little bit above L? A: Both! It shouldn t matter if the y value is bigger or smaller than L. What matters is the distance from y to L.

10 Two important characters ε epsilon and δ delta These two Greek letters are often used in mathematics to represent any small positive real number. By small, we mean close to 0.

11 Definition: We write lim x a f(x) = L if the following holds: for every ε > 0 there exists δ > 0 such that if 0 < x a < δ then f(x) L < ε. This is saying for any ε > 0, there exists a δ > 0 (which depends on ε) such that whenever x is within δ from a, then f(x) will be within ε from L. Note that x a since 0 < x a.

12 lim f(x) = 5 x 3

13 Let f(x) = and let us now prove that { 2x 1 if x 3 6 if x = 3 lim f(x) = 5 x 3

14 Look at Example 2 in the textbook to see how to prove that lim x a f(x) = L for f(x) a linear function. Note that we do not cover limit proofs for functions like those in Example 3 and Example 4. We only cover linear functions.

15 Understanding the formal definition Tools to enrich calculus Click on Limits and Derivatives and then Precise Definitions of Limits. Move ε to a value greater than zero and then see what δ value gives you an interval (a δ, a + δ) for which the distance from f(x) to L will always be less than ε (i.e. f(x) L < ε). Note that x (a δ, a + δ) x a < δ.

16 Yesterday we stated eleven different Limit Laws but we did not prove any of them. We couldn t prove them because we didn t know how. With our new formal definition of a Limit, we can prove the Limit Laws. Proofs of the Limit Laws can be found in Appendix F. These are not examinable in this course, but please take a look at them if you are keen to have a better understanding of the ε-δ definition of a Limit.

17 Infinite Limits Previously we said that f(x) has an infinite limit as x a if we can make f(x) as big as we like by taking x close to a. Now we want to formalise this in the same way that we formalised the definition of a (non-infinite) limit.

18 Definition: Let f be a function defined on some open interval that contains a, except possibly at a itself. Then lim f(x) = x a means that for every positive number M there is a δ > 0 such that if 0 < x a < δ then f(x) > M.

19 Example: f(x) = 1 x 2

20 Definition: Let f be a function defined on some open interval that contains a, except possibly at a itself. Then lim f(x) = x a means that for every negative number N there is a δ > 0 such that if 0 < x a < δ then f(x) < N.

21 Continuity (Ch 2.5)

22 Definition: A function f is continuous at a number a R if lim x a f(x) = f(a). Very important: the above definition implies there are three conditions that must be satisfied for a function to be continuous at a: (1) f must be defined at a (2) lim x a f(x) must exist (3) lim x a f(x) = f(a)

23 If a function f(x) is not continuous when x = a, we say that f is discontinuous at a. A point at which a function is discontinuous is called a discontinuity.

24 Examples of discontinuities

25 More examples of discontinuities

26 Examples: Where are the following discontinuous? (a) f(x) = x2 x 2 { x 2 1 if x 0 (b) f(x) = x 2 1 if x = 0 { x 2 x 2 x 2 if x 2 (c) f(x) = 1 if x = 2

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28 Types of discontinuities In example (a) and example (c) we have a removable discontinuity. We call them removable because if we redefined that point we could remove the discontinuity. The discontinuity in (b) is called an infinite discontinuity.

29 Another example (d) Consider the greatest integer function x. x = the largest integer that is less than or equal to x Examples: 0.5 = = 2 3 = 3 Where is this function discontinuous?

30 Greatest integer function f(x) = x Example (d) has jump discontinuities.

31 One-sided continuity: Definition: A function f is continuous from the right at a if lim f(x) = f(a) x a + and continuous from the left at a if lim f(x) = f(a) x a

32 Example of one-sided continuity: The Heaviside function: { 0 if t < 0 H(t) = 1 if t 0 The function is continuous from the right at 0, but not continuous from the left at 0. This is because but lim H(t) = 1 = H(0) t 0 + lim H(t) = 0 H(0) t 0

33 Definition: A function f is continuous on an interval if it is continuous at every number in the interval. Note: if f is defined only on one side of an endpoint of the interval, we understand continuous to mean continuous from the right or continuous from the left.

34 Example: Show that f(x) = 1 1 x 2 is continuous on the interval [ 1, 1]. To do this we must prove three equalities: lim f(x) = f(a) for any a ( 1, 1) x a lim f(x) = f( 1) x 1 + lim f(x) = f(1). x 1

35 Building continuous functions: Theorem: if f and g are continuous at a and c is a constant, then the following functions are also continuous at a: 1. f + g 2. f g 3. cf 4. fg 5. f g if g(a) 0

36 Proving (1): If f and g are continuous at a then the function f + g is also continuous at a. Q: How do we prove this? A: Using Limit Laws and the definition of the function f + g. In particular we use the fact that the limit of a sum is the sum of the limits (provided that both limits exist).

37 Theorem: (a) Any polynomial is continuous everywhere; that is, it is continuous on R = (, ). (b) Any rational function is continuous wherever it is defined; that is, it is continuous on its domain.

38 Theorem: the following types of functions are continuous at every number in their domains: polynomials rational functions root functions trig functions inverse trig functions exponential functions logarithmic functions

39 Example: f(x) = 2x + 8a if x < 0 ax 2 4b if 0 x 2 3x + 18 if x > 2 What values of a and b make f continuous everywhere?

40 Khan Academy exercises Continuity exercises

41 Example: Where and why is the following function continuous: f(x) = ln(x) + arctan(x) x 2 1