Chapter Goals: Evaluate limits. Evaluate one-sided limits. Understand the concepts of continuity and differentiability and their relationship.

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1 MA123, Chapter 3: The idea of its (pp ) Date: Chapter Goals: Evaluate its. Evaluate one-sided its. Understand the concepts of continuit and differentiabilit and their relationship. Assignments: Assignment 4 Assignment 5 Earlier, the idea of its came up naturall in the course of defining the derivative of a function at a point. We now stud its more sstematicall. Computing a it means computing what happens to the value of a function as the variable in the epression gets closer and closer to (but does not equal) a particular value. The basic definition of it: Let f be a function of. The epression f() = L means that as gets closer and closer to c, through values both smaller and larger than c, but not equal to c, then the values of f() get closer and closer to the value L. Note: It ma sometimes happen that the it does not eist. Eample 1: Compute gets close to 2 from the left gets close to 2 from the right It looks like that =. Eample 2: Suppose that, instead of calculating all the values in the above tables, ou simpl substitute the value = 2 into What do ou find? + 2 Note: The method of substituting in the iting value of the variable works because the operations of arithmetic, namel, addition, subtraction, multiplication, and division, all behave reasonabl with respect to the idea of getting closer to as long as nothing illegal happens. The one illegalit ou will mainl have to watch out for is division b zero. More precisel, if f and g are two functions one has: f() + g() = f() + g() f() g() = f() g() f() g() = f() g() f() f() g() = g() as long as g() 25

2 Eample 3: Compute 1 [( ) (2 4)] Eample 4: Compute Some complications with the definition of its: The following eamples are meant to illustrate the fact that the computation of f() does not reduce to the mere substitution of the value of c in place of in the epression defining f(). The unusual functions described in what follows are introduced to emphasize the fact that the notion of it reall involves what happens to the values of f() as gets closer to the fied value c, and not what the value of f() at = c is. In addition, the most interesting its generall arise precisel when substitution gives an illegal epression involving division b, or even an epression of the form. The latter case occurs for eample when computing the derivative of a function. Eample 5: g() = is shown to the right. Compute g(). 1 The graph of the function if 1 3 if = 1 g() Eample 6: The graph of the function 2 3 h() = is shown to the right. if > 2 if Analze h(). 2 26

3 The previous eample shows that the it of a function as the variable approaches a fied value does not necessaril eist. This brings us to the following notions: One-sided its: A one-sided it epresses what happens to the values of an epression as the variable in the epression gets closer and closer to some particular value c from either the left on the number line (that is, through values less than c) or from the right on the number line (that is, through values greater than c). The notation is: f() }{{} it from the left of c f() } + {{} it from the right of c Fact: f() eists if and onl if both f() and f() + eist and have the same value. The problem of division b zero and a finite nonzero numerator: When this happens, it is standard to sa that the epression is getting arbitraril large (in the positive or negative direction) or is going to (positive or negative) infinit, denoted b ±. As infinit is not reall a number, the epression is not reall getting close to an particular real number. Thus, technicall speaking, the it does not eist. Eample 7: Analze 1 5 ( 1) 2. 2 Eample 8: Analze Eample 9: Analze the it. 27

4 The case : The most interesting and important situation with its is when a substitution ields. This is precisel the situation we are confronted with when attempting to compute derivatives from the definition. The result ields absolutel no information about the it. It does not even tell us that the it does not eist. The onl thing it tells us is that we have to do more work to determine the it. 4 Eample 1: Find the it. ( 2 Eample 11: Find the it ) Eample 12: Find the it (h 3) 2 9 Eample 13: Find the it. h h 28

5 Eample 14: Find the its p() Limits at infinit: A function f() is said to be a rational function if it is of the tpe q(), where p() and q() are both polnomials in. Sometimes we are interested in determining the behavior of a rational function for large (positive or negative) values of the variable. This will be the case, for eample, in Chapter 9. There is a general principle that makes computing these its eas. The idea is that, for ver large (positive or negative) values of, the term with the highest power of has the most influence on the behavior of the polnomial. In other words, when is ver large, the term with the highest power dominates the other terms. Theorem: Let p() and q() be polnomials. Then ± Eample 15: Let p() = Find the its p() q() = ± highest order term of p() highest order term of q(). (2 + 1) 2 Eample 16: Find the it Eample 17: Find the it of the sum n as the number of terms in the sum tends to infinit. (Hint: write the sum as a decimal number first.) 29

6 Continuit and differentiabilit: We first give a brief, non-rigorous and intuitive eplanation of two fundamental notions in Calculus whose definitions involve its. We then discuss how these two notions relate to each other. Definition of continuit: A function f is continuous at a point = c if f() = f(c). A function f is continuous on an interval if it is continuous at ever point of that interval. Note: Geometricall, this means that the graph of f has no holes, jumps, or gaps at an point in the domain of f. Thus ou can draw the graph of f from one end of the interval to the other without lifting our pencil off the paper. Fact: If f and g are continuous functions at c then kf(), f() + g(), f() g() and constant f(), where g(c), are continuous at c. g() Eamples: Polnomials are continuous at ever point. Rational functions are continuous at ever point in their domain. 2 3 if 1 Eample 18: Consider the function f() = 2 + B if > 1 Graph the function f when B = 4 and B = 1. Find a value of B such that the function is continuous at = 1. 3

7 Eample 19 (Greatest integer function): Let f be the function that associates to an value of the greatest integer less than or equal to. Compute the values of f at =.5, 1.99, 2, 2.1, 4.87, 1.5. Make a graph of the function f. Compute f() and f() if n < n + 1, n is an odd integer Eample 2: Let f be the function defined as follows f() = if n < n + 1, n is an even integer Sketch the graph of f(). Where is f discontinuous? Definition of differentiabilit: A function f is said to be differentiable at = c if the it f (c) = h f(c + h) f(c) h eists. Note: Geometricall, this means that at an point (,f()) on the graph of f there is a well defined tangent line (so the graph is smooth there, and does not have a sharp point), and furthermore the tangent line is not vertical. Eamples: Polnomials are differentiable at ever point. Rational functions are differentiable at ever point in their domain. 31

8 Let us describe the situations in which a function fails to be differentiable at one point, despite the fact that it is continuous at the point. Eample 21 (Vertical tangent lines): Consider the function f() =. What can ou sa about the tangent line to the graph of f at the point =? Eample 22 (Corner points): line to the graph of f at the point =? Consider the function f() =. What can ou sa about the tangent Theorem: If f is differentiable at = c, then f is also continuous at = c. Equivalentl, if f is not continuous at = c, then f is not differentiable at = c. Note: If f is not differentiable at = c, then f ma or ma not be continuous at = c. (a) f() = is not differentiable at =, but it is continuous at =. (b) f() = 1 is not differentiable at = and it is not continuous at =. Eample 23: Let 2 if 2 f() = m + b if > 2. Find the values of m and b that make f differentiable at = 2. 32

9 The intermediate value theorem and zeros of functions Intermediate Value Theorem (IVT): Suppose that f is continuous on the closed interval [a, b] and let k be an number strictl between f(a) and f(b). Then there is at least one number c in (a,b) such that f(c) = k. f(b) k f(a) a c 1 c 3 c 3 b The following result helps us find zeros of functions (i.e roots). Eistence of Zeros Theorem: Consider the above situation where f(a) and f(b) have opposite signs. f(b) a c 1 c 2 c 3 b f(a) Then b IVT, there eists at least one number c in (a,b) such that f(c) =. Eample 24: Does the equation = have a root inside the interval (,1)? 33