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Calculus Fundamentals

Limits of Functions

In the last quiz, we looked at some examples of limits. Here is the general idea.

We say that the limit of the function ff as xx approaches aa is the number LL if, as xx gets closer and closer to a,a, the function values f(x)f(x) get closer and closer to L.L. If there is no such number L,L, we say the limit does not exist.

When the limit exists, we use the notation lim⁡x→af(x)=L.\lim\limits_{x \to a} f(x) = L.

In this picture, for example, the limit of the function (in blue) as xx approaches 2 (from either side) is 4.

As the input approaches 2, the output approaches 4. As the input approaches 2, the output approaches 4.

In some ways this is a simple idea, but as we’ll see, there are plenty of subtleties involved!

                         
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Limits of Functions

Let’s start with a straightforward example. Here’s a graph of a function f.f. What is the limit of f(x)f(x) as xx approaches 1? In other words, as the input gets closer and closer to 1, what value is the output getting closer to?

                         
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Limits of Functions

Now consider the function ff given by the formula:

f(x)={x2,x≠13,x=1f(x) = \begin{cases}\begin{array}{rl} x^2, & x \neq 1 \\ 3, & x = 1 \\ \end{array}\end{cases}

In other words, ff is the usual function y=x2,y=x^2, except that we’ve set the value at x=1x=1 to be 3. What is lim⁡x→1f(x)\lim\limits_{x \to 1} f(x)?

                         
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Limits of Functions

In the previous example, the value of the function at 1 was 3. But the limit was still 1, because as the xx values get closer and closer to 1, the function values get closer and closer to 1. This is an important point:

lim⁡x→af(x)\lim\limits_{x \to a} f(x) has nothing to do with the value of ff at aa itself! It only says something about what happens as xx gets close to aa.

                         
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Limits of Functions

Here’s another interesting example. Define

f(x)={−1,x≤01,x>0f(x) = \begin{cases}\begin{array}{rl} -1, & x \leq 0 \\ 1, & x > 0 \\ \end{array}\end{cases}

What is lim⁡x→0f(x)\lim\limits_{x \to 0} f(x)?

Note: For this to exist, the value of the function, f(x),f(x), must be getting closer to some number LL as xx gets closer to 0… no matter how close xx gets to 0.

                         
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Limits of Functions

This is our first example in this quiz of a limit that doesn’t exist. It’s true that as xx approaches 0 from the right, the function values approach 1. And as xx approaches 0 from the left, the function values approach -1. But this means there’s no single LL that the function approaches no matter how close xx gets to 0. So the limit doesn’t exist.

This example, where the "right-hand" (as xx approaches from the right) and "left-hand" (as xx approaches from the left) limits exist but aren’t equal, is the simplest way a limit might not exist. But there are many other ways. For example, in the previous quiz we saw that lim⁡x→0sin⁡(1x)\lim\limits_{x \to 0} \sin\left(\frac{1}{x}\right) does not exist, because as xx gets small, 1x\frac{1}{x} gets large, and so sin⁡\sin just oscillates between -1 and 1, instead of approaching any particular L.L.

                         
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Limits of Functions

The function f(x),f(x), shown below, is defined on the interval (0,9].(0,9].

How many of the limits below exist?

  • lim⁡x→2f(x)\displaystyle \lim_{x \rightarrow 2} f(x)

  • lim⁡x→3f(x)\displaystyle \lim_{x \rightarrow 3} f(x)

  • lim⁡x→5f(x)\displaystyle \lim_{x \rightarrow 5} f(x)

  • lim⁡x→7f(x)\displaystyle \lim_{x \rightarrow 7} f(x)

                         
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Limits of Functions

Usually when we need to compute a limit in calculus, we won’t be presented with a graph, but with an algebraic expression. For example, let

f(x)=x2+2x−8x−2f(x) = \frac{x^2+2x-8}{x-2}

What is lim⁡x→3f(x)?\lim\limits_{x \to 3} f(x)?

                         
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Limits of Functions

The last example was easy, because everything was well-behaved at x=3.x=3. (In a later quiz, we’ll see this happens whenever the function is continuous.) Now consider:

lim⁡x→2x2+2x−8x−2\lim_{x \to 2} \frac{x^2+2x-8}{x-2}

The function is undefined at x=2,x=2, because of the denominator. We simply cannot evaluate f(2).f(2). But we can still investigate the limit as xx approaches 2, because that only depends on what ff is doing near 2, not at 2. In fact, notice that the numerator is also 0 when you plug in 2. This is another example of a 00\frac{0}{0} indeterminate form from the first chapter. When we encounter such a thing, the limit is not obvious. Often though, we can discover it by algebraic manipulation.

What is the limit? (Hint: factor the numerator.)

                         
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Limits of Functions

For the last three questions, we’re going to look at a strange and interesting example. The same basic idea behind limits hasn’t changed: lim⁡x→af(x)=L\lim\limits_{x \to a} f(x) = L means that as xx approaches a,a, the values f(x)f(x) approach L.L.

Define:

f(x)={x,if   x=1n where n is an integer 0,otherwise.f(x) = \begin{cases}\begin{array}{rl} x, & \text{if } \; x = \frac{1}{n} \text{ where } n \text{ is an integer } \\ 0, & \text{otherwise.} \\ \end{array}\end{cases}

For example, f(12)=12,f(\frac{1}{2}) = \frac{1}{2}, f(0)=0,f(0) = 0, and f(23)=0.f(\frac{2}{3}) = 0.

Try to get a feel for what this function looks like. There will be a picture on the next page, but see if you can work it out without looking.

                         
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Limits of Functions

Let's look at a function ff that is 0 almost everywhere except for 1n\frac{1}{n} points, which lie on the line y=xy=x.

The graph of ff looks something like this:

Now, what is the limit of f(x)f(x) as xx approaches 37\frac{3}{7}?

                         
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Limits of Functions

The value of \(f\) is 0, except for at \(\frac{1}{n}\) points, which lie on the line \(y=x\). The value of ff is 0, except for at 1n\frac{1}{n} points, which lie on the line y=xy=x.

What about lim⁡x→13f(x) ?\displaystyle\lim_{x \to \frac{1}{3}} f(x)\ ?

                         
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Limits of Functions

The value of \(f\) is 0, except for at \(\frac{1}{n}\) points, which lie on the line \(y=x\). The value of ff is 0, except for at 1n\frac{1}{n} points, which lie on the line y=xy=x.

Now for the most interesting question. What is lim⁡x→0f(x) ?\lim\limits_{x \to 0} f(x)\ ?

It’s hard to visualize exactly what’s happening near 0. But you know the rule: the function value is 0, except at points like 12,\frac{1}{2}, 13,\frac{1}{3}, 14,\frac{1}{4}, etc. So as xx gets smaller and smaller, what happens to the f(x)f(x) values?

                         
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