Mathematics 242 – Principles of Analysis Discussion 4 – Limits for f(x), a First Approach September 29, 2000

Background

Our next goal is to study precisely what it should mean to say that a real-valued function of a real variable x has a limit as x → a. That is we want to understand what should it mean to say that lim_{x→a} f(x) = L for some real number L. (This will be used to define the derivative of a function so it’s really the basis for calculus.) For the moment, we will have to rely on the intuitive idea of limits of functions that you probably used in your calculus courses. Namely, saying that lim_{x→a} f(x) = L should mean roughly that the values of the function f(x) are approaching L as x approaches a (but x ≠ a). You should be aware, though, that you have not seen a very broad class of examples of functions, so your intuition may not be too reliable(!)

Fortunately, we do know a lot about limits of sequences now, so it is natural to try to use this here to capture the x → a part of the limit idea. Today we will try to determine exactly what the connection should be. Note that if we have a sequence x_{n} → a and all the x_{n} are in the domain of the function, then we can substitute these x-values into f, and get a corresponding sequence of function values f(x_{n}), and we can ask what the sequence f(x_{n}) is doing as n → ∞.

Discussion Questions

A) Consider the function f(x) = x^{2} + 2 and a = 3.

1) Based on the intuitive idea, should lim_{x→3} f(x) exist? What should it be?

2) Let x_{n} = 3+\frac{1}{n}, a sequence converging to a = 3. What can you say about lim_{n→∞} f(x_{n})? (Use the “Big Limit Theorem” here and whenever possible in the following. It is not necessary to use the ϵ − n_{0} definition of convergence for sequences.)

3) Same question for y_{n} = \frac{3n^{2}−1}{n^{2}+1} → 3.

4) (You’ll need a calculator for this one.) Experiment with the sequence z_{n} = 9n sin (\frac{1}{3n}). What does it seem to be doing as n → ∞? What does the sequence f(z_{n}) seem to be doing as n → ∞?

B) Now consider the function

f(x) = \{ \begin{matrix}x^{2} & if x ≠ 0 \\ 1 & if x = 0 \end{matrix}

1) Based on the intuitive idea, should lim_{x→0} f(x) exist? What should it be?

2) Take the sequence x_{n} = \frac{(−1)^{n}}{n^{2}} → 0. Does f(x_{n}) have a limit as n → ∞?

3) What about y_{n} = \frac{10^{n}}{n!} → 0? What can you say about f(y_{n}) as n → ∞? (You will probably want to refer back to Discussion 3, question B here.)

4) (Careful, careful!) Consider the sequence z_{n} = \frac{(1+(−1)^{n})}{2^{n}}. What does this sequence do as n → ∞? What does f(z_{n}) do as n → ∞?

C) Now consider the following function:

f(x) = \{ \begin{matrix}1 & if x is an irrational number \\ \frac{1}{p} & if x = \frac{m}{p^{k}} in lowest terms, p prime \\ \frac{1}{m} & if x = \frac{m}{n} is rational, but n not a prime power \end{matrix}

For instance,

f(\sqrt{2}) = 0 since \sqrt{2} is irrational
f(\frac{3}{64}) = \frac{1}{2} since 64 = 2^{6} is a power of a prime
f(\frac{17}{100}) = \frac{1}{17} since 100 = 2^{2}5^{2} is not a power of a prime

It’s not so easy to apply the intuitive idea in a case like this, so let’s try to use some sequences converging to 0 to try to see whether lim_{x→0} f(x) should exist.

1) Consider the sequence x_{n} = \frac{\sqrt{2}}{5^{n}} → 0. Does f(x_{n}) have a limit as n → ∞?

2) What about f(y_{n}) for y_{n} = \frac{1}{7^{n}} → 0?

3) What about f(z_{n}) for z_{n} = \frac{(2+(−1)^{n})}{n} → 0?

4) What do these examples tell you about the possibility of identifying a real number L such that lim_{x→0} f(x) = L for this function?

D) Putting it all together. Write a paragraph addressing the following questions: If you want to try to determine whether lim_{x→a} f(x) = L, how can you use sequences x_{n} → a? When should it be true that lim_{x→a} f(x) exists and equals L? (Be careful here; the example in B, part 4 is important!) When would lim_{x→a} f(x) not exist? Explain how each part of the previous three questions fits into what you are saying.

Assignment

Group writeups, due in class Wednesday, October 4.