1. Chapter 0ne Limits and Rates of Change up down return end

2. 1.4 The Precise Definition of a Limit We know that it means f(x) is moving close to L while x is moving close to a as we desire. And it can reaches L as near as we like only on condition of the x is in a neighbor. (2) DEFINITION Let f(x) be a function defined on some open interval that contains the number a, except possibly at a itself. Then we say that the limit of f(x) as x approaches a is L, and we write, if for very number  >0 there is a corresponding number  >0 such that |f(x) - L|<  whenever 0<|x - a|< . up down return end How to give mathematical description of

3. In the definition, the main part is that for arbitrarily  >0, there exists a  >0 such that if all x that 0<|x - a|<  then |f(x) - L|< . Another notation for is f(x) L as x a. Geometric interpretation of limits can be given in terms of the graph of the function y=L+y=L+ y=L -  y=Ly=L a a-  a+  ox y=f(x)y=f(x)y up down return end

4. Example 1 Prove that Solution Let  be a given positive number, we want to find a positive number  such that |(4x-5)-7|<  whenever 0<|x-3|< . But |(4x-5)-7|=4|x-3|. Therefore 4|x-3|<  whenever 0<|x-3|< . That is, |x-3|<  /4 whenever 0<|x-3|< . Example 2 Prove that Example 3 Prove that up down return end

5. Example 4 Prove that Similarly we can give the definitions of one-sided limits precisely. (4)DEFINITION OF LEFT-SIDED LIMIT If for every number  >0 there is a corresponding number  >0 such that |f(x) - L|<  whenever 0< a - x < , i.e, a -  < x < a. (5)DEFINITION OF LEFT-SIDED LIMIT If for every number  >0 there is a corresponding number  >0 such that |f(x) - L|<  whenever 0< x - a < , i.e, a < x < a + . Example 5 Prove that up down return end

6. Example 6 If prove that (6) DEFINITION Let f(x) be a function defined on some open interval that contains the number a, except possibly at a itself. Then we say that the limit of f(x) as x approaches a is infinity, and we write, if for very number M>0 there is a corresponding number  >0 such that f(x)>M whenever 0< |x - a|< . up down return end

7. Example Prove that Example 5 Prove that (6)DEFINITION Let f(x) be a function defined on some open interval that contains the number a, except possibly at a itself. Then we say that the limit of f(x) as x approaches a is infinity, and we write, if for very number N 0 such that f(x)<N whenever 0< |x - a|< . up down return end

8. Similarly, we can give the definitions of one-side infinite limits. Example Prove that up down return end

9. 1.5 Continuity If f(x) not continuous at a, we say f(x) is discontinuous at a, or f(x) has a discontinuity at a. (1) Definition A function f(x) is continuous at a number a if. (3) A function f(x) is continuous at a number a if and only if for every number  >0 there is a corresponding number  >0 such that |f(x) - f(a) |<  whenever |x - a|< . Note that: (1) f(a) is defined (2) exists. up down return end

10. Example is discontinuous at x=2, since f(2) is not defined. Example is continuous at x=2.. Example Prove that sinx is continuous at x=a. (2) Definition A function f(x) is continuous from the right at every number a if A function f(x) is continuous from the left at every number a if up down return end

11. (2) Definition A function f(x) is continuous on an interval if it is continuous at every number in the interval. (at an endpoint of the interval we understand continuous to mean continuous from the right or continuous from the left) Example At each integer n, the function f(x)=[x] is continuous from the right and discontinuous from the left. Example Show that the function f(x)=1-(1-x 2 ) 1/2 is continuous on the interval [-1,1]. (4)Theorem If functions f(x), g(x) is continuous at a and c is a constant, then the following functions are continuous at a: 1. f(x)+g(x) 2. f(x)-g(x) 3. f(x)g(x) 4. f(x)[g(x)] -1 (g(a) isn’t 0.) up down return end

12. (5) THEOREM (a) any polynomial is continuous everywhere, that is, it is continuous on R 1 =(  ). (b) any rational function is continuous wherever it is defined, that is, it is continuous on its domain. Example Find (6) THEOREM If n is a positive even integer, then f(x)= is continuous on [0,  ). If n is a positive odd integer, then f(x)= is continuous on (  ). Example On what intervals is each function continuous? up down return end

13. (8) THEOREM If g(x) is continuous at a and f(x) is continuous at g(a) then (f o g)(x))= f(g(x)) is continuous at a. (7) THE INTERMEDIATE VALUE THEOREM Suppose that f(x) is continuous on the closed interval [a,b]. Let N be any number strictly between f(a) and f(b). Then there exists a number c in (a,b) such that f(c)=N y x b y=Ny=N a (7) THEOREM If f(x) is continuous at b and, then up down return end

14. Example Show that there is a root of the equation 4x 3 - 6x 2 + 3x -2 =0 between 1 and 2. up down return end

15. 1.6 Tangent, and Other Rates of Change A. Tangent (1) Definition The Tangent line to the curve y=f(x) at point P( a, f(a)) is the line through P with slope provided that this limit exists. Example Find the equation of the tangent line to the parabola y=x 2 at the point P(1,1). up down return end

16. B. Other rates of change The difference quotient is called the average rate change of y with respect x over the interval [x 1, x 2 ]. (4) instantaneous rate of change= at point P(x 1, f(x 1 )) with respect to x. Suppose y is a quantity that depends on another quantity x. Thus y is a function of x and we write y=f(x). If x changes from x 1 and x 2, then the change in x (also called the increment of x) is  x= x 2 - x 1 and the corresponding change in y is  x= f(x 2 ) - f(x 1 ). up down return end

17. (1) what is a tangent to a circle? Can we copy the definition of the tangent to a circle by replacing circle by curve? 1.1 The tangent and velocity problems The tangent to a circle is a line which intersects the circle once and only once. How to give the definition of tangent line to a curve? For example, up down return end

18. Fig. (a) In Fig. (b) there are straight lines which touch the given curve, but they seem to be different from the tangent to the circle. L2 L2 Fig. (b) L1L1 up down return end

19. Let us see the tangent to a circle as a moving line to a certain line: So we can think the tangent to a curve is the line approached by moving secant lines. P Q up down return end Q'Q'

20. x m PQ 2 3 1.5 2.5 1.1 2.1 1.01 2.01 1.001 2.001 Example 1: Find the equation of the tangent line to a parabola y=x 2 at point (1,1). Q is a point on the curve. Q y=x2 y=x2 P up down return end

21. Then we can say that the slope m of the tangent line is the limit of the slopes m QP of the secants lines. And we express this symbolically by writing And So we can guess that slope of the tangent to the parabola at (1,1) is very closed to 2, actually it is 2. Then the equation of the tangent line to the parabola is y-1=2(x-2) i.e y=2x-3. up down return end

22. Suppose that a ball is dropped from the upper observation deck of the Oriental Pearl Tower in Shanghai, 280m above the ground. Find the velocity of the ball after 5 seconds. From physics we know that the distance fallen after t seconds is denoted by s(t) and measured in meters, so we have s(t)=4.9t 2. How to find the velocity at t=5? (2) The velocity problem: Solution up down return end

23. So we can approximate the desired quantity by computing the average velocity over the brief time interval of the n-th of a second from t=5, such as, the tenth, twenty-th and so on. Then we have the table: Time interval Average velocity(m/s) 5<t<6 53.9 5<t<5.1 49.49 5<t<5.05 49.245 5<t<5.01 49.049 5<t<5.001 49.0049

24. The above table shows us the results of similar calculations of average velocity over successively smaller time periods. It also appears that as time period tends to 0, the average velocity is becoming closer to 49. So the instantaneous velocity at t=5 is defined to be the limiting value of these average velocities over shorter time periods that start at t=5. up down return end

25. 1.2 The Limit of a Function Let us investigate the behavior of the function y=f(x)=x 2 -x+2 for values of x near 2. x f(x) 1.0 2.000000 3.0 8.000000 1.5 2.750000 2.5 5.750000 1.8 3.440000 2.2 4.640000 1.9 3.710000 2.1 4.310000 1.95 3.852500 2.05 4.152500 1.99 3.970100 2.01 4.030100 1.995 3.985025 2.001 4.003001 up down return end

26. We see that when x is close to 2(x>2 or x<2), f(x) is close to 4. Then we can say that: the limit of the function f(x)=x 2 -x+2 as x approaches 2 is equal to 4. Then we give a notation for this : In general, the following notation:

27. (1) Definition: We write Guess the value of. Notice that the function is not defined at x=1, and x 1 f(x) 0.5 0.666667 1.5 0.400000 0.9 0.526316 1.1 0.476190 0.99 0.502513 1.01 0.497512 0.999 0.500250 1.001 0.499750 0.999. 0.500025 1.0001 0.499975 Example 1 up down return end and say “the limit of f(x), as x approaches a, equals L”. Solution If we can make the values of f(x) arbitrarily close to L (as close to L as we like)by taking x to be sufficiently close to a but not equal to a. Sometimes we use notation f(x) L as x a.

28. Example 1 Find Example 2 Find Notice that as x a which means that x approaches a, x may >a and x may <a. Example 3 Discuss, where The function H(x) approaches 0 as x approaches 0 and x<0, and it approaches 1, as x approaches 0 and x>0. So we can not say H(x) approaches a number as x a. up down return end

29. One -side Limits: Even though there is no single number that H(x) approaches as t approaches 0. that is, does not exist. But as t approaches 0 from left, t<0, H(x) approaches 0. Then we can indicate this situation symbolically by writing: But as t approaches 0 from right, t>0, H(x) approaches 1. Then we can indicate this situation symbolically by writing: up down return end

30. We write And say the left-hand limit of f(x) as x approaches a (or the limit of f(x) as x approaches a from left) is equal to L. That is, we can make the value of f(x) arbitrarily close to L by taking x to be sufficiently close to a and x less than a. And say the right-hand limit of f(x) as x approaches a (or the limit of f(x) as x approaches a from right) is equal to L. That is, we can make the value of f(x) arbitrarily close to L by taking x to be sufficiently close to a and x greater than a. We write Here x a + ” means that x approaches a and x>a. (2)Definition: Here x a - ” means that x approaches a and x<a. up down return end

31. See following Figure: What will it happen as x a or x b? x Oab y=f(x)y=f(x) y up down return end

32. (3)Theorem:if and only if Example: Find. x 1/x 2 ±1 1 ±0.5 4 ±0.2 25 ±0.1 100 ±0,05 400 ±0,01 10000 ±0.001 1000000 x y=1/x 2 O y up down return end

33. To indicate the kind of behavior exhibited in this example, we use the notation: Generally we can give following Example Find The another notation for this is f(x) as x a, which is read as “the limit of f(x), as x approaches a, is infinity” or “f(x) becomes infinity as x approaches a” or “f(x) increases without bound as x approaches a”. (4)DEFINITION: Let f be a function on both sides of a, except possibly at a itself. Then means that values of f(x) can be made arbitrarily large ( as we please) by taking x sufficiently close to a ( but not equal to a). up down return end

34. y=f(x)=ln|x| y x Obviously f(x)=ln|x| becomes large negative as x gets close to 0. (5)DEFINITION: Let f be a function on both sides of a, except possibly at a itself.Then means that values of f(x) can be made arbitrarily large ( as we please) by taking x sufficiently close to a ( but not equal to a). The another notation for this is f(x) - as x a, which is read as “the limit of f(x), as x approaches a, is negative infinity” or “f(x) becomes negative infinite as x approaches a” or “f(x) decreases without bound as x approaches a”. up down return end

35. (6)DEFINITION: The line x=a is called a vertical asymptote of the curve y=f(x) if at least one of the following statements is true: Similar definitions can be given for one-side infinite limits. Remember the meanings of x a - and x a +. up down return end

36. FindExampleand up down return end

37. 1.3 Calculating limits using limit laws LIMIT LAWS Suppose that c is a constant and the limits exist. Then and 1. 2. 3. 5. 4 up down return end

38. (if n is even,we assume that ) 6. where n is a positive integer, 7. 8. 9 where n is a positive integer, 10. where n is a positive integer, (if n is even, we assume that a>0) 11. where n is a positive integer, up down return end

39. Example 6. Calculate Example 1. Find Example 2. Find Example 3. Calculate Example 4. Calculate Example 5. Calculate where up down return end

40. If f(x) is a polynomial or rational function and a is in the domain of f(x), then (1) THEOREM if and only if Example : Show that Example: If,determine whether exists. Example: Prove thatdoes not exists. Example: Prove thatdoes not exists, where value of [x] is defined as the largest integer that is less than or equal to x. up down return end

41. (2) THEOREM If f(x) g(x) for all x in an open interval that contains a (except possibly at a) and the limits of f and g exist as x approaches a, then (3)SQUEEZE THEOREM If f(x) g(x) h(x) for all x in an open interval that contains a (except possibly at a) and then Example: Show that up down return end