Index FAQ Functional limits Intuitive notion Definition How to calculate Precize definition Continuous functions

Index FAQ Intuitive meaning of the limit of a function You write, which means that as x “approaches” c, the function f( x) “approaches” the real number L

Index FAQ Video help: Lesson 1: What Is a Limit? Lesson 2: When Does a Limit Exist? Lesson 3: How do you evaluate limits? Worked out EXAMPLES: Intuitive meaning of the limit of a function

Index FAQ Intuitive meaning of the limit of a function The limit of a function f( x) is a number, what the function intends to take, what we can observe om the graph of it in other word: the number to which the functional values approach either in the infinity, or negative infinity, or at a certain point, which NOT necessarily belongs to the domain of the function. The limit might or might not be equal to the functional value at that point in which the limit is taken

Index FAQ Limit of the Function Note: we can approach a limit from left … right …both sides Function may or may not exist at that point At a right hand limit, no left function not defined At b left handed limit, no right function defined a b

Index FAQ Intuitive meaning of the limit of a function

Index FAQ Intuitive meaning of the limit of a function You write : which means that as x “approaches” c, the function f( x) “approaches” the real number L

Index FAQ Can be observed on a graph. Observing a Limit

Index FAQ Observing a Limit Can be observed on a graph.

Index FAQ Non Existent Limits f(x) grows without bound

Index FAQ Intuitive meaning of the limit of a function

Index FAQ Intuitive meaning of the limit of a function What is the number this function does intend to take? -In the infinity? -In the negative infinity? -At zero? - At x=1 from the right? - At x=1 from the left? -At 2?

Index FAQ What is the number this function does intend to take? -In the infinity: -In the negative infinity? -At zero? -At x=1 from the right? At x=1 from the left? At x=2, substituting 2: Intuitive meaning of the limit of a function

Index FAQ Intuitive meaning of the limit of a function What function could it be?

Index FAQ Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Computing Limits: substitution Ex.

Index FAQ Non Existent Limits

Index FAQ Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. One-Sided Limit of a Function The right-hand limit of f (x), as x approaches a, equals L written: if we can make the value f (x) arbitrarily close to L by taking x to be sufficiently close to the right of a. a L

Index FAQ Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. One-Sided Limit of a Function The right-hand limit of f (x), as x approaches a, equals L written: if we can make the value f (x) arbitrarily close to L by taking x to be sufficiently close to the right of a. a L

Index FAQ Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. One-Sided Limit of a Function The left-hand limit of f (x), as x approaches a, equals M written: if we can make the value f (x) arbitrarily close to L by taking x to be sufficiently close to the left of a. a M

Index FAQ One sided limits Numbers x near c fall into two natural categories: those that lie to the left of c and those that lie to the right of c. We write [The left-hand limit of f(x) as x tends to c is L.] to indicate that as x approaches c from the left, f(x) approaches L. We write [The right-hand limit of f(x) as x tends to c is L.] to indicate that as x approaches c from the right, f(x) approaches L For a full limit to exist, both one-sided limits have to exist and they have to be equal.

Index FAQ Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. One-Sided Limit of a Function Ex. Given Find The limit does not exists at 3, but it exists from the left, and from the right

Index FAQ 1, if x ＞ 0 −1, if x ＜ 0. Let’s try to apply the limit process at different numbers c. If c ＜ 0, then for all x sufficiently close to c, x ＜ 0 and f(x) = −1. It follows that for c ＜ 0 lim f(x) = lim (−1) = −1 x → c If c ＞ 0, then for all x sufficiently close to c, x ＞ 0 and f(x) = 1. It follows that for c ＜ 0 lim f(x) = lim (1) = 1 x → c However, the function has no limit as x tends to 0: lim f(x) = −1 but lim f(x) = 1. x → 0 - x → 0 + Example

Index FAQ Computing Limits We saw already the first step: substitution If fails: try to factorize the terms, then simplify

Index FAQ 24 Example Graph it. Substitotion failed, but the limit exist!!!! What happens at x = 2?

Index FAQ Good job if you saw this as “limit does not exist” indicating a vertical asymptote at x = -2. This limit is indeterminate. With some algebraic manipulation, the zero factors could cancel and reveal a real number as a limit. In this case, factoring leads to…… The limit exists as x approaches 2 even though the function does not exist. In the first case, zero in the denominator led to a vertical asymptote; in the second case the zeros cancelled out and the limit reveals a hole in the graph at (2, ¼). Examples

Index FAQ Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Computing Limits Ex Note: f (-2) = 1 is not involved The limiot exists at -2 because the left and right hand limits are equal

Index FAQ Computing Limits We saw already the first step: substitution If fails: try to factorize the terms, then simplify If fails: try the „conjugate”

Index FAQ Using Conjugates

Index FAQ Find: When we graph this function, the limit appears to be zero. so for : by the sandwich/squeeze/pinching theorem:

Index FAQ Well-known limits xx 5. x f(x) a

Index FAQ Well-known... If r>1 Can you tell, what if r<1? 6.

Index FAQ Squeezing/pinching/sandwich theorem for functions Suppose that g(x)  h(x)  f(x), in the neirbourhood of x=c, (not necesserily at c though) and lim g(x)=limf(x)=L at x=c Then lim h(x) exists at c, and lim(h(x))=L

Index FAQ Proof We are considering the area of triangle OAB, circle section OAB and triangle OAD

Index FAQ The limit of sin(x)/x as x goes to 0 is proof Since applying squeeze theorem

Index FAQ Proof of 22

Index FAQ Applying well-known limits Example 1 Find Solution To calculate the first limit, we “pair off” sin 4x with 4x and use (2.5.6): Therefore, The second limit can be obtained the same way:

Index FAQ Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Limit of a Function The limit of f (x), as x approaches a, equals Lwritten: if we can make the value f (x) arbitrarily close to L by taking x to be sufficiently close to a. a L

Index FAQ Formal Definition of a Limit If for any  (as close as you want to get to L) there exists a  (we can get as close as necessary to c), such as: Then the limit exits:

Index FAQ Formal Definition of a Limit The For any ε (as close as you want to get to L) There exists a  (we can get as close as necessary to c )

Index FAQ Specified Epsilon, Required Delta

Index FAQ Formal Definition of a Limit If for any  (as close as you want to get to L) there exists a  (we can get as close as necessary to c), such as: Then the limit exits:

Index FAQ Finding the Required  Consider showing |f(x) – L| = |2x – 7 – 1| = |2x – 8| <  We seek a  such that when |x – 4| <  |2x – 8|< for any  we choose It can be seen that the  we need is

Index FAQ No limit Example Here we set f(x) = sin (π/ x) and show that the function can have no limit as x → 0 The function is not defined at x = 0, as you know, that’s irrelevant. What keeps f from having a limit as x → 0 is indicated in the Figure above As x → 0, f(x) keeps oscillating between y = 1 and y = –1 and therefore cannot remain close to any one number L.

Index FAQ Finite limit in the infinity Definition: The limit of a function in the infinity is A if for arbitrary  >0 there exists a positive number K, such that if x>K, then  f(x)-A  <  Example: f(x)=sinx/x+A

Index FAQ Infinit limit at a (finite) point Definition: Let x 0 is a point of the domain of the definition of function f. The limiting value of f at x 0 is (positive) infinity, if for all K>0 there exists a  >0 such that if  x-x 0  K Example: 1/x at 0

Index FAQ Limiting value – defitions HAND IN!! B ased on the previous defitinions, define the following: -Limit in the infinity is infinity/negative infinity -Limit in the negatíve infinity is + infinity/-infinity -Limim in the negative infintiy is a number A

Index FAQ Continuity Continuity at a Point The basic idea is as follows: We are given a function f and a number c. We calculate (if we can) both and f (c). If these two numbers are equal, we say that f is continuous at c. Here is the definition formally stated.

Index FAQ Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Continuity of a Function A function f is continuous at the point x = a if the following are true: a f(a)f(a)

Index FAQ Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Continuous Functions A polynomial function y = P(x) is continuous at every point x. A rational function is continuous at every point x in its domain. All elementary functions and their inverses are continuous If f and g are continuous at x = a, then

Index FAQ 50 f (x) = (x 2 – 9)/(x + 3) at x = -3 c. b. - 6 The limit exist! f (-3) = 0/0 a. Is undefined! Therefore the function is not continuous at x =

Index FAQ A function f is said to be continuous on an interval if it is continuous at each interior point of the interval and one-sidedly continuous at whatever endpoints the interval may contain. For example: The function is continuous on [−1, 1] because it is continuous at each point of (−1, 1), continuous from the right at −1, and continuous from the left at 1. The graph of the function is the semicircle.. Continuity on Intervals

Index FAQ Classification of points of discontinuity:first and second kind Infinite (dicont. of second kind): at least one of the one sides limits does not exists, or functional values tend to the infinity/- infinity Removable: limit from the right and from the left exist, and equal Jump: Limits exist from both side, but not equal Discontinuity of first kind: removable, jump

Index FAQ 53 Important Theorems about continuous functions Extreme Value Theorem Weirerstrass, r ETV Intermediate Value Theorem - ITV Some applications

Index FAQ Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Intermediate Value Theorem If f is a continuous function on a closed interval [a, b] and L is any number between f (a) and f (b), then there is at least one number c in [a, b] such that f(c) = L. ab f (a) f (b) L c f (c) =

Index FAQ Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Intermediate Value Theorem Ex. f (x) is continuous (polynomial) and since f (1) 0, by the Intermediate Value Theorem there exists a c on [1, 2] such that f (c) = 0.

Index FAQ 56 Limitations of IVT The IVT is a powerful tool, but it has its limitations. To illustrate, suppose that d(t) represents the decibel level of Pork Chop's motorcycle engine, and suppose d(0) = 100 and d(10) = 35, where t is measured in seconds. d is a continuous function. By IVT in the ten second interval between time t=0 and time t=35 Pork Chop's decibel level reached every value between 35 and 100. It does NOT say anything about: When or how many times (other than at least once) a particular decibel was attained. Whether or not decibel levels bigger than 100 or less than 35 were reached.

Index FAQ 57 Definition of absolute extrema Suppose that f is a function defined on a domain D containing c. Then Absolute maximum value at c if f(c)  f(x) for all x  D Absolute minimum value at c if f(c)  f(x) for all x  D

Index FAQ 58 Extreme value theorem WEIERSTRASS OR EVT Can find absolute extrema under certain hypotheses: If f is continuous on a closed interval [a,b], with -  < a < b < , then f has an absolute maximum M and an absolute minimum m on [a,b]

Index FAQ 59 Example No maximum or minimum value on the domain. However, on [-3,3], it has both. Question: does function f fullfil EVT?

Index FAQ 60 Conclusions about hypotheses Conclude that hypothesis that interval be closed, [a,b], essential Conclusion that f is continuous also essential:

Index FAQ 61 Examples fulfilling hypotheses f(x) = 2 - 3x where -5 < x < 8 g(x) = sin(x) where 0 < x < 2 

Index FAQ 62 Limitations of Extreme Value Theorem Polynomial f(x)=x 5 - 3x is continuous everywhere Must have absolute max, min on [-1, 10] by theorem Theorem doesn’t say where these occur Extreme value theorem just an “existence theorem” Learn tools for finding extrema later using the derivative