Limits of Functions f(x) (a, L) L f(x) x a x x 20
Informal Definition: If the values of can be made as close to as we like by taking values of sufficiently close to [but not equal to ] then we write or as 21
Observe: " " means can approach from either side On a sketch, the graph of approaches the 2-D plane location [destination] called,, but the graph itself may have no point, occupying that location! may not be The language to describe how the outputs behave as the inputs approaches number 22
Example: We examine the graph? 1 1 Domain of 1 1 0, 0 1 0, 1 1, 0 23
4 3,5 3 2,5 2 1,5 1 0,5 0-2 -1 0 1 2 3 4 5 6 Conjecture: 2 1 1 24
General definition: Let be a function and a real number (that may be or may be not in the domain of ). We say that the it as approaches of is L, written if can be made arbitrarily close to by choosing sufficiently close to (but not equal to). If no such number exists, then we say that does not exist. Warning: Not all its exist! 25
Example: 1 0 1 0 1 1 0 from the left, 1 0 from the right, 1 So has no meaning! 26
Two-Sided and One-Sided Limits Notation approaches from the left [minus in a superscript position] or [comes up to ] or approaches from the right [plus in a superscript position] or [comes down to ] or 27
Relationship between Two-Sided and One-Side Limits: Theorem [if and only if]: exists exists and both equal 28
Example: 1 does not exist 1 29
The Algebra of Limits as Basic Limits as,, used in polynomial functions and rational functions. The constant function The identity function: The reciprocal ( flip over ) function: 1 1 30
Limits of Sums, Differences, Products, Quotients and Roots The Rules of Algebra for Limits Let be any real number and then 31
Provided 0 L Provided when = even then 0 32
Limits of Polynomial Function Polynomial Expressions A monomial (one-term polynomial) has the form A real number constant called a coefficient Subscript is a label Two monomial with the same degree and same variable are called like terms : ; - like terms A polynomial in one variable has the standard form: [higher powers lower powers] 0 leading coefficient 33 - a variable n=0,1,2, 3, not negative called the degree of the monomial
By the Rules of Algebra for Limits we can break down polynomials into simpler parts Example: 4 3 4 3 4 3 5 4 5 3 8 34
For any polynomial function For polynomial, this it is the same as substitution of for 35
Limits of Rational Functions and the appearance of There are 3 cases to consider Case 1: 0 Limit Example: 4 5 3 2 5 4 3 5 2 4 2 3 44 36
Case 2: 0 0 Limit does not exist (division by 0!) Classic Examples: 1 1 37
a) 1 b) 1 38
Case 3: 0 0 Limit determine whether the it exists or not, without more work! Example: 4 2 0 2 2 an indeterminate form: We cannot 2 2 2 4 2 This is only one particularly technique! Does not work always! 39
The Algebra of Limits as : End Behavior Basic Limits: The constant function and The identity function: 40
The reciprocal ( flip over ) function: 1 0 1 0 41
Limits of Sums, Differences, Products, Quotient and Roots The Rules of Algebra for Limits applied to or We only state for case As before, suppose: then 42
L 1 1 0 Provided 0 Provided when = even then 0 43
A polynomial function Where 0 Limits of Polynomial Functions: Two End Behaviors The two end behaviors are that as (the rightward end) or (the leftward end) Then Observe: The two possibilities! 44
So, the end behavior of matches the end behavior of Theorem: Example: 4 17 3 2 50 4 45
Limits of Rational Functions: Three Types of End Behavior The Degree of a polynomial is the exponent of the highest power of in the polynomial 46
Type 1. Deg(top)=Deg(bottom) Example: 7 4 7 4 1 7 1 7.... 47
Type 2. Deg(top)<Deg(bottom) Example: Always zero 0 5 2 2 1 5 2 2 1 2 0 0 the (-axis) is a horizontal asymptote 48
Type 3. Deg(top)>Deg(bottom) If 0 If 0 Always one of these Always one of these Example: 4 5 1 4 5 1 1 49
Limits of and note, that makes no sense 0 0 50
Limits of Trigonometric Functions 0 1 0 0 2 2 51
Some more techniques for computing its a) Rational functions: divide the top and the bottom to cancel (reduce): factor and cancel 3 2 1 2 1 2 2 1 1 2 1 1 2 1 1 3 2 52
b) If we have some roots: expand the top and the bottom with a factor: The standard thing to do with a square root in a sum or difference is rationalize 1 2 4 5 1 2 1 2 4 5 1 2 5 1 5 1 2 1 1 1 2 1 4 53
c) One-sided its:,, Substitution,, replace and by 0 2 1 9 : 3 3 0 23 1 6 2 1 9 3 9 9 6 54 7 2 6 2 1 9 : 3 3 0
Finding a Limit by Squeezing Old problem: How do we calculate Answer: we squeeze Theorem [ squeezing Theorem] 0 If for all in some open interval containing and too. Then 55
Idea: h(c) g(c) c 56
? 57 1
Area 1 2 1 2 1 1 2 1 2 1 1 1 58
Since The squeeze Theorem implies, that 0 1 1 1 1 59
Exercise 1 1 1 1 1 1 1 1 1 1 0 1 1 0 60
Exercise 2 7 3 7 3 7 7 7 7 3 1 7 3 3 7 3 1 1 1 7 3 61
Continuous Function Continuous Function at a single point What properties of a function cause breach or holes in the graph? (so, it does not continue ) Definition: A function is continuous at provided all three conditions are satisfied. 1. is defined [ exists at ] 2. exists [equal a real number] 3. If not, the is discontinuous at 62
63
Intuitively, is continuous at if the Graph of does not break at If is not continuous at (i.e. if the Graph of does break at ), then is a discontinuity of Note: If is an endpoint for the Domain of, then in the definition is replaced by the appropriate one-sided it, e.g. Is defined on 0, and is continuous at 0 because 0 0 and 0 64
Function Continuous on an Interval is continuous on, or, if is continuous at each in the interval. [two-sided its are possible here at each ] What about defined on, [No two-sided its at or ] Definition: at is continuous from the left, if from the right if if 65
Definition: is continuous on,, if 1. is continuous on, 2. is continuous from the right at 3. is continuous from the left at 66
Properties and combinations of continuous functions Recall: If is a polynomial function, then So, every polynomial function is continuous everywhere. Suppose,, are continuous at Theorem: ; ; are all also continuous at is continuous at provided 0 [otherwise, is discontinuous at ] So, every rational function is continuous at every point where the bottom is not zero. 67
Lastly: the composition of continuous functions is also continuous Theorem: If is continuous at and is continuous at then is continuous at 68
The Intermediate Value Theorem and Approximating Roots: Intermediate Value Theorem If is continuous on, and is between and, or equal to one of them, then there is at least one value of in, such that 69
Theorem: If is continuous on, and, are non zero with opposite signs, then there is at least one solution of 0 in, 70