The Limit Process THE LIMIT PROCESS (AN INTUITIVE INTRODUCTION) We could begin by saying that limits are important in calculus, but that would be a major understatement. Without limits, calculus would not exist. Every single notion of calculus is a limit in one sense or another. For example: What is the slope of a curve? It is the limit of slopes of secant lines. What is the length of a curve? It is the limit of the lengths of polygonal paths inscribed in the curve.
The Limit Process What is the area of a region bounded by a curve? It is the limit of the sum of areas of approximating rectangles.
The Limit Process The Idea of a Limit We start with a number c and a function f defined at all numbers x near c but not necessarily at c itself. In any case, whether or not f is defined at c and, if so, how is totally irrelevant. Now let L be some real number. We say that the limit of f (x) as x tends to c is L and write lim f x c ( x) = L provided that (roughly speaking) as x approaches c, f(x) approaches L or (somewhat more precisely) provided that f (x) is close to L for all x c which are close to c.
The Limit Process Example Set f ( x) = 1 x and take c = 8. As x approaches 8, 1 x approaches 9 and that lim f x = 3 x 8 ( ) 1 x approaches 3. We conclude If for that same function we try to calculate lim x 2 f ( x) ( ) 1 we run into a problem. The function f x = x is defined only for x 1. It is therefore not defined for x near 2, and the idea of taking the limit as x approaches 2 makes no sense at all: lim x 2 f ( x) does not exist.
The Limit Process The curve in Figure 2.1.4 represents the graph of a function f. The number c is on the x-axis and the limit L is on the y-axis. As x approaches c along the x-axis, f (x) approaches L along the y-axis.
The Limit Process As we have tried to emphasize, in taking the limit of a function f as x tends to c, it does not matter whether f is defined at c and, if so, how it is defined there. The only thing that matters is the values taken on by f at numbers x near c. Take a look at the three cases depicted in Figure 2.1.5. In the first case, f (c) = L. In the second case, f is not defined at c. In the third case, f is defined at c, but f (c) L. However, in each case lim x c because, as suggested in the figures, ( ) f x = L as x approaches c, f (x) approaches L.
The Limit Process Example Set f x ( ) 2 x 9 = x 3 and let c = 3. Note that the function f is not defined at 3: at 3, both numerator and denominator are 0. But that doesn t matter. For x 3, and therefore for all x near 3, ( x )( x+ ) x 2 9 3 3 = = x + 3 x 3 x 3 2 x 9 Therefore, if x is close to 3, then = x + 3 x 3 is close to 3 + 3 = 6. We conclude that 2 x 9 lim = lim( x + 3) = 6 x 3 x 3 x 3
The Limit Process 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 to indicate that We write to indicate that lim x c ( ) f x = L [The left-hand limit of f(x) as x tends to c is L.] as x approaches c from the left, f(x) approaches L. lim x c + ( ) f x = L [The right-hand limit of f(x) as x tends to c is L.] as x approaches c from the right, f(x) approaches L
The Limit Process Example Take the function indicated in Figure 2.1.7. As x approaches 5 from the left, f (x) approaches 2; therefore x 5 ( x) lim f = 2 As x approaches 5 from the right, f (x) approaches 4; therefore ( ) x 5 ( x) lim f = 4 + The full limit, lim f x, does not exist: consideration of x < 5 would force the x 5 limit to be 2, but consideration of x > 5 would force the limit to be 4. For a full limit to exist, both one-sided limits have to exist and they have to be equal.
The Limit Process Example For the function f indicated in figure 2.1.8, In this case ( ) ( ) It does not matter that f ( 2) = 3. Examining the graph of f near x = 4, we find that Since these one-sided limits are different, ( ) x 2 x 2 ( ) lim f x = 5 and lim f x = 5 + x 2 ( x) lim f = 5 ( ) ( ) lim f x = 7 whereas lim f x = 2 + x 4 x 4 lim x 4 f ( x) does not exist.
The Limit Process Remark To indicate that f (x) becomes arbitrarily large, we can write f (x). To indicate that f (x) becomes arbitrarily large negative, we can write f (x). Consider Figure 2.1.10, and note that for the function depicted there the following statements hold: + as x 3, f (x) ( ) and as x 3, f (x). Consequently, Also, + as x 3, f (x). as x 7, f (x) and as x 7, f (x). We can therefore write as x 7, f (x).
The Limit Process Summary of Limits That Fail to Exist Examples 7-13 illustrate various ways in which the limit of a function f at a number c may fail to exist. We summarize the typical cases here: ( ) ( ) (i) lim f x = L (Examples 7, 8). 1, lim f x = L2 and L1 L + 2 x c x c (The left-hand and right-hand limits of f at c each exist, but they are not equal.) (ii) f(x) + as x c, or f(x) + as x c +, or both (Examples 9, 10, 11). (The function f is unbounded as x approaches c from the left, or from the right, or both.) (iii) f(x) oscillates as x c, c + or c (Examples 12, 13).
Definition of Limit
Definition of Limit Figures 2.2.1 and 2.2.2 illustrate this definition.
Definition of Limit In Figure 2.2.3, we give two choices of ε and for each we display a suitable δ. For a δ to be suitable, all points within δ of c (with the possible exception of c itself) must be taken by the function f to within ε of L. In part (b) of the figure, we began with a smaller ε and had to use a smaller δ. The δ of Figure 2.2.4 is too large for the given ε. In particular, the points marked x 1 and x 2 in the figure are not taken by f to within ε of L.
Definition of Limit The limit process can be described entirely in terms of open intervals as shown in Figure 2.2.5.
Definition of Limit For each number c For each real number c For each constant k
Definition of Limit There are several different ways of formulating the same limit statement. Sometimes one formulation is more convenient, sometimes another, In particular, it is useful to recognize that the following four statements are equivalent:
Definition of Limit One-sided limits give us a simple way of determining whether or not a (two-sided) limit exists:
Definition of Limit Example For the function defined by setting ( ) lim f x x 0 f ( x) = 2 does not exist. 2x+ 1, x 0 x x, x> 0 Proof The left- and right-hand limits at 0 are as follows: 2 ( ) ( ) ( ) ( ) lim f x = lim 2x + 1 = 1, lim f x = lim x x = 0 + + x 0 x 0 x 0 x 0 Since these one-sided limits are different, ( ) lim f x x 0 does not exist.
Limit Theorems
Limit Theorems The following properties are extensions of Theorem 2.3.2.
Limit Theorems Examples 3x 5 6 5 1 lim = = x 2 2 x + 1 4+ 1 5 3 2 x 3x 27 27 lim = = 0 x 3 2 1 x 1 9
Limit Theorems Examples From Theorem 2.3.10 you can see that 2 x 3x 7 5 lim lim lim x 1 2 2 x 1 x x 4 x 0 x All fail to exist.
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 lim f ( x and f (c). If these two numbers are equal, we x ) c say that f is continuous at c. Here is the definition formally stated. If the domain of f contains an interval (c p, c + p), then f can fail to be continuous at c for only one of two reasons: either (i) f has a limit as x tends to c, but lim f ( x) f ( c), or x c (ii) f has no limit as x tends to c. In case (i) the number c is called a removable discontinuity. The discontinuity can be removed by redefining f at c. If the limit is L, redefine f at c to be L. In case (ii) the number c is called an essential discontinuity. You can change the value of f at a billion points in any way you like. The discontinuity will remain.
Continuity The functions shown have essential discontinuities at c. The discontinuity in Figure 2.4.2 is, for obvious reasons, called a jump discontinuity. The functions of Figure 2.4.3 have infinite discontinuities.
Continuity
Continuity Example The function ( ) F x 3 x x = 3 x + + 4 2 x 5x+ 6 is continuous at all real numbers other than 2 and 3. You can see this by noting that where F = 3 f + g/h + k f (x) = x, g(x) = x 3 x, h(x) = x 2 5x + 6, k(x) = 4. Since f, g, h, k are everywhere continuous, F is continuous except at 2 and 3, the numbers at which h takes on the value 0. (At those numbers F is not defined.)
Continuity
Continuity
Continuity Continuity on Intervals 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: (i) The function f x = 1 x ( ) 2 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. (ii) The function 1 f ( x) = 2 is continuous on ( 1, 11) x because it is continuous at each point of ( 1, 1). It is not continuous on [ 1, 1) because it is not continuous from the right at 1. It is not continuous on ( 1, 1] because it is not continuous from the left at 1. (iii) The function graphed in Figure 2.4.8 is continuous on (, 1] and continuous on (1, ). It is not continuous on [1, ) because it is not continuous from the right at 1. (iv) Polynomials, being everywhere continuous, are continuous on (, ). Continuous functions have special properties not shared by other functions.
Trigonometric Limits
Trigonometric Limits From this it follows readily that
Trigonometric Limits
Trigonometric Limits In more general terms, Example Find sin 4x 1 cos 2x lim and lim x 0 3x x 0 5x Solution To calculate the first limit, we pair off sin 4x with 4x and use (2.5.6): Therefore, sin 4x 4 sin 4x 4 sin 4x 4 4 lim = lim lim ( 1) x 0 3x x 0 3 4x = = = 3 x 0 4x 3 3 The second limit can be obtained the same way: 1 cos 2x 2 1 cos 2x 2 1 cos 2x 2 lim = lim = lim = ( 0) = 0 x 0 5x x 05 2x 5 x 0 2x 5
Two Basic Theorems A function which is continuous on an interval does not skip any values, and thus its graph is an unbroken curve. There are no holes in it and no jumps. This idea is expressed coherently by the intermediate-value theorem.
Two Basic Theorems Boundedness; Extreme Values A function f is said to be bounded or unbounded on a set I in the sense in which the set of values taken on by f on the set I is bounded or unbounded. For example, the sine and cosine functions are bounded on (, ): 1 sin x 1 and 1 cos x 1 for all x (, ). Both functions map (, ) onto [ 1, 1]. The situation is markedly different in the case of the tangent. (See Figure 2.6.4.) The tangent function is bounded on [0, π/4]; on [0, π/2) it is bounded below but not bounded above; on ( π/2, 0] it is bounded above but not bounded below; on ( π/2, π/2) it is unbounded both below and above.
Two Basic Theorems For a function continuous on a bounded closed interval, the existence of both a maximum value and a minimum value is guaranteed. The following theorem is fundamental.
Two Basic Theorems From the intermediate-value theorem we know that continuous functions map intervals onto intervals. Now that we have the extreme-value theorem, we know that continuous functions map bounded closed intervals [a, b] onto bounded closed intervals [m, M]. Of course, if f is constant, then M = m and the interval [m, M] collapses to a point.