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Our aim in this section is to explore the meaning of the limit of a function. We begin by showing how the idea of a limit arises when we try to find the velocity of a falling ball. Our aim in this section is to explore the meaning of the limit of a function. We begin by showing how the idea of a limit arises when we try to find the velocity of a falling ball.
SECTION 1.3 THE LIMIT OF A FUNCTION THE LIMIT OF A FUNCTION Our aim in this section is to explore the meaning of the limit of a function We begin by showing how the idea of a limit arises when we try to find the velocity of a falling ball 1.3 P2 Example Suppose that a ball is dropped from the upper observation deck of the CN Tower in Toronto, 450 m above the ground Find the velocity of the ball after seconds 1.3 P3 Example SOLUTION Through experiments carried out four centuries ago, Galileo discovered that the distance fallen by any freely falling body is proportional to the square of the time it has been falling Remember, this model neglects air resistance If the distance fallen after t seconds is denoted by s(t) and measured in meters, then Galileo’s law is expressed by the following equation s(t) = 4.9t2 1.3 P4 Example SOLUTION The difficulty in finding the velocity after s is that you are dealing with a single instant of time (t = 5) No time interval is involved However, we can approximate the desired quantity by computing the average velocity over the brief time interval of a tenth of a second (from t = to t = 5.1) 1.3 P5 Example SOLUTION change in position average velocity = time elapsed s ( 5.1) − s ( ) = 0.1 4.9 ( 5.1) − 4.9 ( ) = 0.1 = 49.49 m/s 2 1.3 P6 Example SOLUTION The table shows the results of similar calculations of the average velocity over successively smaller time periods It appears that, as we shorten the time period, the average velocity is becoming closer to 49 m/s 1.3 P7 Example SOLUTION The instantaneous velocity when t = is defined to be the limiting value of these average velocities over shorter and shorter time periods that start at t = Thus, the (instantaneous) velocity after s is: v = 49 m/s 1.3 P8 INTUITIVE DEFINITION OF A LIMIT Let’s investigate the behavior of the function f defined by f(x) = x2 – x + for values of x near The following table gives values of f(x) for values of x close to 2, but not equal to 1.3 P9 INTUITIVE DEFINITION OF A LIMIT From the table and the graph of f (a parabola) shown in Figure 1, we see that, when x is close to (on either side of 2), f(x) is close to 1.3 P10 Example Find lim if it exists x →0 x SOLUTION As x becomes close to 0, x2 also becomes close to 0, and 1/x2 becomes very large 1.3 P46 Example SOLUTION In fact, it appears from the graph of the function f(x) = 1/x2 that the values of f(x) can be made arbitrarily large by taking x close enough to Thus, the values of f(x) not approach a number So, lim x →0 does not exist x 1.3 P47 PRECISE DEFINITION OF A LIMIT Definition is appropriate for an intuitive understanding of limits, but for deeper understanding and rigorous proofs we need to be more precise 1.3 P48 PRECISE DEFINITION OF A LIMIT We want to express, in a quantitative manner, that f(x) can be made arbitrarily close to L by taking to x be sufficiently close to a (but x ≠ a This means f(x) that can be made to lie within any preassigned distance from L (traditionally denoted by ε, the Greek letter epsilon) by requiring that x be within a specified distance δ (the Greek letter delta) from a 1.3 P49 PRECISE DEFINITION OF A LIMIT That is, | f(x) – L | < ε when | x – a | < δ and x ≠ a Notice that we can stipulate that x ≠ a by writing < | x – a | The resulting precise definition of a limit is as follows 1.3 P50 Definition Let f be a function defined on some open interval that contains the number , except possibly at a itself Then we say that the limit of f(x) as x approaches a is L, and we write lim f ( x) = L x →a if for every number ε > there is a corresponding number δ > such that if < | x – a | < δ then | f(x) – L | < ε 1.3 P51 PRECISE DEFINITION OF A LIMIT If a number ε > is given, then we draw the horizontal lines y = L + ε and y = L – ε and the graph of f 1.3 P52 PRECISE DEFINITION OF A LIMIT f ( x) = L , then we can find a number δ > If lim x→a such that if we restrict x to lie in the interval (a – δ) and (a + δ) take x ≠ a, then the curve y = f(x) lies between the lines y = L – ε and y = L + ε 1.3 P53 PRECISE DEFINITION OF A LIMIT It’s important to realize that the process illustrated in Figures 12 and 13 must work for every positive number ε, no matter how small it is chosen Figure 14 shows that if a smaller ε is chosen, then a smaller δ may be required 1.3 P54 PRECISE DEFINITION OF A LIMIT In proving limit statements it may be helpful to think of the definition of limit as a challenge First it challenges you with a number ε Then you must be able to produce a suitable δ You have to be able to this for every ε > 0, not just a particular 1.3 P55 Example x − 5)=7 Prove that lim(4 x →3 SOLUTION Let ε be a given positive number According to Definition with a = and L = 7, we need to find a number δ such that if < | x – | < δ then | (4x – 5) – | < ε 1.3 P56 Example SOLUTION But |(4x – 5) – 7| = |4x – 12| = |4(x – 3)| = 4|(x – 3)| Therefore, we want: if < | x – | < δ then 4|(x – 3)| < ε We can choose δ to be ε/4 because if < | x – | < δ = ε/4 then 4|(x – 3)| < ε Therefore, by the definition x − 5) − of a limit, lim(4 x →3 1.3 P57 PRECISE DEFINITION OF A LIMIT For a left-hand limit we restrict x so that x < a, so in Definition we replace < | x – a | < δ by x – δ < x < a Similarly, for a right-hand limit we use a < x < a + δ 1.3 P58 Example 10 Prove that lim x = x → 0+ SOLUTION Let ε be a given positive number We want to find a number δ such that if < x < δ then x −0 [...]... calculator’s value for t 2 + 9 is 3.000… to as many digits as the calculator is capable of carrying 1.3 P22 THE LIMIT OF A FUNCTION Something very similar happens when we try to graph the function t +9 −3 f ( t) = t2 of the example on a graphing calculator or computer 2 1.3 P23 THE LIMIT OF A FUNCTION These figures show quite accurate graphs of f and, when we use the trace mode (if available), we can estimate... x a is f ( x) → L as x → a which is usually read “f(x) approaches L as x approaches a. ” 1.3 P13 THE LIMIT OF A FUNCTION Notice the phrase “but x ≠ a in the definition of limit This means that, in finding the limit of f(x) as x approaches a, we never consider x = a In fact, f(x) need not even be defined when x = a The only thing that matters is how f is defined near a 1.3 P14 THE LIMIT OF A FUNCTION. .. x) says that we consider values of definition of lim x a x that are close to a but not equal to a 1.3 P16 Example 2 SOLUTION The tables give values of f(x) (correct to six decimal places) for values of x that approach 1 (but are not equal to 1) On the basis of the values, we make the guess that x −1 = 0.5 2 x →1 x − 1 lim 1.3 P17 THE LIMIT OF A FUNCTION Example 2 is illustrated by the graph of f... 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 (on either side of a) but not equal to a 1.3 P12 THE LIMIT OF A FUNCTION Roughly speaking, this says that the values of f(x) tend to get closer and closer to the number L as x gets closer and closer to the number a (from either side of a) but x ≠ a An alternative notation for... values but, eventually, you will get the value 0 if you make t sufficiently small 1.3 P21 THE LIMIT OF A FUNCTION Does this mean that the answer is really 0 instead of 1/6? No, the value of the limit is 1/6, as we will show in the next section The problem is that the calculator gave false values because t 2 + 9 is very close to 3 when t is small In fact, when t is sufficiently small, a calculator’s... of t near 0 As t approaches 0, the values of the function seem to approach 0.16666666… So, we guess that: t2 + 9 − 3 1 lim = 2 t →0 6 t 1.3 P20 THE LIMIT OF A FUNCTION What would have happened if we had taken even smaller values of t? The table shows the results from one calculator You can see that something strange seems to be happening If you try these calculations on your own calculator, you... DEFINITION OF A LIMIT In fact, it appears that we can make the values of f(x) as close as we like to 4 by taking x sufficiently close to 2 We express this by saying the limit of the function f(x) = x2 – x + 2 as x approaches 2 is equal to 4.” The notation for this is: lim ( x 2 − x + 2 ) = 4 x→2 1.3 P11 Definition 1 We write lim f ( x ) = L x a and say the limit of f(x), as x approaches a, equals L”... easily that the limit is about 1/6 1.3 P24 THE LIMIT OF A FUNCTION However, if we zoom in too much, then we get inaccurate graphs—again because of problems with subtraction 1.3 P25 Example 4 sin x Guess the value of lim x →0 x SOLUTION The function f(x) = (sin x)/x is not defined when x = 0 Using a calculator (and remembering that, if x ∈ , sin x means the sine of the angle whose radian measure... sin nπ = 0 for any integer n, it is also true that f(x) = 1 for infinitely many values of x that approach 0 1.3 P30 Example 5 SOLUTION The graph of f is given in Figure 7 The dashed lines near the y-axis indicate that the values of sin(π/x) oscillate between 1 and –1 infinitely as x approaches 0 1.3 P31 Example 5 SOLUTION Since the values of f(x) do not approach a fixed number as approaches 0, π lim... P32 THE LIMIT OF A FUNCTION Examples 3 and 5 illustrate some of the pitfalls in guessing the value of a limit It is easy to guess the wrong value if we use inappropriate values of x, but it is difficult to know when to stop calculating values As the discussion after Example 3 shows, sometimes, calculators and computers give the wrong values In the next section, however, we will develop foolproof