Calculus: What is a Limit? (understanding epislon-delta proofs)

Transcription

1 Calculus: What is a Limit? (understanding epislon-delta proofs) Here is the definition of a limit: Suppose f is a function. We say that Lim aa ff() = LL if for every εε > 0 there is a δδ > 0 so that if 0 < aa < δδ then ff() LL < εε It may look complicated but if we break it down piece by piece we ll be able to understand it. x a Let s begin with the most important part of the definition, the x a part. To understand x a it s helpful to imagine motion. (For many people, this is a new meaning of x. They have seen x represent a number as in x + 3 = 8. And they ve seen x represent any number as in x + 5 = 5 + x. But here x is a changing quantity.) For instance, imagine x is a car that starts at 0 and then moves to 0.9 then 0.99, then 0.999, and continues like this forever. We can ask some questions that all mean the same thing. What number is x approaching? What number is x getting closer and closer to? What number is x converging to? The questions all have the same answer. x is approaching the number 1

2 x is getting closer and closer to the number 1 x is converging to the number 1 Or, in mathematical notation, x 1 So x 1 means x is approaching 1, x is getting closer and closer to 1, x is converging to 1. More precisely, it means for any small number, for instance , that eventually x is within of 1. In other words, it means that eventually x 1 < is true. And it means it is true not only for but for any small number we can name. If we symbolize any small number we can name as δδ, then we can say x approaches 1 x gets closer and closer to 1 x converges to 1 x 1 all mean that eventually the following is true x 1 < δδ for any δδ we choose. Notice that x may never reach 1, but it does get closer and closer to 1 that is, it does converge to 1 and that is all we need to say x 1. But what if it does reach 1? What if x takes on these values? { 0, 0.9, 1, 1, 1, 1... } That case isn t interesting or useful for our purposes. So we d like to exclude it. How? Simple, we insist 0 < x 1 < δδ which is why we see the following in the limit definition. There s another type of convergence we should mention, convergence to infinity. It s written as x and it means that eventually x gets larger than any positive number we choose. For instance, we d say that {1, 2, 3, 4,... } converges to infinity.

3 Summary: x 1 means we have an infinite sequence of numbers that gets closer and closer to 1, in the sense that 1 x is eventually less than any small number we can name. But x never becomes 1. In other words, 0 < 1 x < δδ for all positive δδ. x means we have an infinite sequence of numbers that gets larger and larger in the sense that x is eventually larger than any number N we can name. (We can also say x if x is eventually smaller than any number N.) f(x) Whether we have x a for some finite number or we have x, it s true that x is the independent variable. So if we are given some function f, we can calculate the value of the dependent variable f(x). And if x converges, we can calculate what the dependent variable f(x) converges to (if it does converge). For example: consider what happens to the the function ff() = 1 as x 0. As x goes to 0, the dependent variable f(x) goes to. x f(x) = 1/x 1 10,000 = 1/ ,000,000 = 1/ ,000,000,000 = 1/ So we can write as x 0, f(x) As another example: consider what happens to the the function ff() = 1 as x. As the independent variable x goes to, the dependent variable f(x) goes to 0. x ,000 1,000,000 1,000,000,000 f(x) = 1/x 0.01 = 1/ = 1/ = 1/ = 1/ So we can write as x, f(x) 0 Of course, convergence need not involve infinity. For example, if f(x) = 2x + 1 x x -> f(x) = 2x f(x) -> Then we can write as x 1, f(x) 3 But in each case the function definition, along with what x converges to, determines what the dependent variable f(x) converges to.

4 The limit So now we know what the limit means. The expression means the same thing as as x 1, f(x) 3 or if x 1, then f(x) 3 So Lim aa ff() = LL " means x a, f(x) L, which means that as the independent variable x gets closer and closer to the number a, the dependent variable f(x) gets closer and closer to the number L. So we ve explained every part of the limit definition except for one point. Here s the definition again. Suppose f is a function. We say that Lim aa ff() = LL if for every εε > 0 there is a δδ > 0 so that if 0 < aa < δδ then ff() LL < εε The point we haven t explained yet is, if we have 0 < 1 x < δδ, then shouldn t we write 0 < ff() LL < εε too? No. If we had that then, for example, for the constant function f(x) = 3 we wouldn t be able to write Another way to think of it is that f(x) is the dependent variable, so controlling the independent variable with 0 < 1 x < δδ is sufficient.

5 Using the limit defintion In beginning Calculus, understanding the limit definition is important. Using the definition to prove f(x) sin converges can involve complicated algebra and even geometry (as in the proof of Lim 0 = 1). So students are often asked to use the defintion to prove only simple cases. Here s an example. Suppose for f(x) = 2x + 1 we want to prove which, as we know, simply means as x 1, f(x) 3. We begin by surveying what we know. We know (because we are given it) that 1. Which means we know 0 < 1 x < δδ for all positive δδ. What we need to prove is that given any positive εε that (2 + 1) 3 < εε Step 1: Write what we want to prove ff() 3 < εε Step 2: Use the function s defintion to relate εε to δδ. ff() 3 < εε (2 + 1) 3 < εε 2 2 < εε 2* 1 < εε 1 < εε/2 Step 3: Read off the proof from the bottom up. That is, for any εε, if we select a δδ = εε/2, we ve proven that ff()) 3 < εε So if we want (2 + 1) 3 to be less than εε, we merely set δδ = εε/2, as the following shows for 3 cases. Given ϵ, we set δ, so x is between and f(x) is between In each case, setting δδ = εε/2 ensures that (2 + 1) 3 < εε. So we ve satisfied the requirements of the limit defintion and proven As we mentioned, the calculations for other functions may be more involved. But the principles and understanding remain unchanged. Dec 4, 2017