In Section 1A, we saw how to go from a position graph to a velocity graph. However, the graphs we were dealing with were piecewise linear, which made it very easy to find the velocities, or the slopes. If the position graphs are not piecewise linear, it is more difficult to find the slope at a given point on the graph.
There is a very nice way of doing this for many functions, but if we’re not careful, it will require division by zero! What does being careful entail? It means knowing your limits!
Numerical Limits
Suppose you wanted to evaluate the function [latex]f(x) = \frac{x^3 - 8}{x - 2}[/latex] at [latex]x = 2[/latex]. Plugging in [latex]x = 2[/latex] into the [latex]f(x)[/latex] formula gives
[latex]\frac{2^3 - 8}{2 - 2} = \frac{0}{0}[/latex]
But there is a problem — we’ve divided by zero.
(image credit: Jaggery)
Hopefully you didn’t actually do the division by zero. What can we do instead? Let’s make a table to see what happens when we get close to putting in two, without actually doing it.
| x | f(x) |
|---|---|
| 1.5 | 9.25 |
| 1.9 | 11.41 |
| 1.99 | 11.9401 |
| 1.999 | 11.994 |
| 1.9999 | 11.9994 |
| 2.0001 | 12.0006 |
| 2.001 | 12.006 |
| 2.01 | 12.0601 |
| 2.1 | 12.61 |
| 2.5 | 15.25 |
If you look at the table, it looks like [latex]f(x)[/latex] SHOULD be equal to [latex]12[/latex] at [latex]x = 2[/latex]. We can’t plug in [latex]x = 2[/latex] because of division by zero, but it really should be [latex]12[/latex] if it has a value. Can we just say it’s 12 and call it a day? Well, not quite, since we want to distinguish between functions that are actually equal to 12, and ones that just should be. That’s where limits come in.
We say [latex]f(2)[/latex] is undefined, but we can write [latex]\lim_{x \to 2} f(x) = 12[/latex], which we read as “the limit of [latex]f[/latex] of [latex]x[/latex] as [latex]x[/latex] approaches two is twelve”. That example is the idea behind a limit.
More technically, a limit is a value [latex]L[/latex] the [latex]y[/latex]-value of a function [latex]f(x)[/latex] approaches as the [latex]x[/latex]-value approaches a certain value [latex]a[/latex], either from the right, the left or both. Notationally, [latex]\lim_{x \to a^-} f(x) = L[/latex] is the left hand limit, [latex]\lim_{x \to a^+} f(x) = L[/latex] is the right hand limit, and [latex]\lim_{x \to a} f(x) = L[/latex] is the two-sided limit. Let’s look at some pictures to make this more intuitive.
Graphical Limits
Consider the following [latex]f(x)[/latex].
To find the value of a function, recall that you look at how high that function is at a given [latex]x[/latex] value. For example, [latex]f(1)[/latex] is about [latex]2[/latex], and [latex]f(3)[/latex] is about [latex]10[/latex].
What’s happening at [latex]x = 2[/latex], or [latex]f(2)[/latex]? The filled-in circle shows the function value. The white circle indicates that value is not part of the function. So [latex]f(2) = 8[/latex].
However, there is something funny going on at [latex]x = 2[/latex]. Namely, the function seems to “jump from [latex]4[/latex], stop at [latex]8[/latex] momentarily, then finally jump to [latex]12[/latex] and continue. This is called a discontinuity, and is usually a bad thing, or at least something that can be hard to deal with. This is where limits can be helpful. The notation [latex]\lim_{x \to 2^-} f(x)[/latex] indicates the [latex]y[/latex]-value of the function as [latex]x[/latex] approaches the value of [latex]2[/latex] from the left. Here is the picture:
In other words, [latex]\lim_{x \to 2^-} f(x)[/latex] is the value that [latex]f(x)[/latex] should be at [latex]2[/latex], if you were approaching from the left. In this case, [latex]f(x)[/latex] should be 4 if everything were right in the world. Therefore, [latex]\lim_{x \to 2^-} f(x) = 4[/latex]. The limit allows us to fill in what the function should be, even though it isn’t the case.
The right handed limit [latex]\lim_{x \to 2^+} f(x)[/latex] is the same way, but it approaches from the right.
Here, the function approaches [latex]12[/latex] as we approach [latex]2[/latex] from the right, and therefore we write [latex]\lim_{x \to 2^+} = 12[/latex].
A two-sided limit exists if [latex]\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)[/latex]. In other words, if the left and right limits are the same. The notation for this is [latex]\lim_{x \to a} f(x)[/latex]. Since the left and right limits are different, so we just write “Does Not Exist” or “DNE”. So [latex]\lim_{x \to 2} f(x) = \text{DNE}[/latex].
Other Examples
For each value of [latex]a[/latex], find [latex]f(a)[/latex], [latex]\lim_{x \to a^-} f(x)[/latex], [latex]\lim_{x \to a^+} f(x)[/latex], [latex]\lim_{x \to a} f(x)[/latex].
- [latex]a = -5[/latex]
- [latex]a = -3[/latex]
- [latex]a = -1[/latex]
- [latex]a = 1[/latex]
- [latex]a = 3[/latex]
- [latex]a = 5[/latex]
- [latex]a = \infty[/latex]
- [latex]a = -\infty[/latex]
- We are looking at the function at [latex]x = -5[/latex]. The actual value here is [latex]f(-5) = 3[/latex]. However, the value it should be if we were coming at [latex]-5[/latex] from the left is [latex]-2[/latex], so [latex]\lim_{x \to -5^-} f(x) = -2[/latex]. Coming from the right, it should be [latex]3[/latex], so [latex]\lim_{x \to -5^+} f(x) = 3[/latex]. Finally, [latex]\lim_{x \to -5} f(x) = \text{DNE}[/latex], since the left and right limits are not equal.
- We are looking at the function at [latex]x = -3[/latex]. We have [latex]f(-3) = -1[/latex], since that’s where the black dot is. But the value SHOULD be [latex]2[/latex], whether we approach from the right or left. So [latex]\lim_{x \to -3^-} f(x) = \lim_{x \to -3^+} f(x) = 2[/latex]. Since the left and right limits are equal, we have [latex]\lim_{x \to -3} f(x) = 2[/latex].
- We have [latex]f(-1) = 5[/latex], [latex]\lim_{x \to -1^-} f(x) = 4[/latex], [latex]\lim_{x \to -1^+} f(x) = 1[/latex], and [latex]\lim_{x \to -1} f(x) = \text{DNE}[/latex].
- Everything is nice and happy — there are no discontinuities here. Therefore [latex]f(1) = \lim_{x \to 1^-} f(x) = \lim_{x \to 1^+} f(x) = \lim_{x \to 1} f(x) = 0[/latex].
- We have what is called a vertical asymptote, and the function basically goes off infinitely far in both directions. Here, we say [latex]f(3) = \text{DNE}[/latex], since there is no obvious value to make [latex]f(3)[/latex] equal to. We say [latex]\lim_{x \to 3^-} f(x) = \infty[/latex], since it goes up infinitely high (the [latex]\infty[/latex] symbol means “infinity”). Likewise, [latex]\lim_{x \to 3^+} f(x) = -\infty[/latex], since it goes down infinitely low. Therefore, we can see [latex]\lim_{x \to 3} f(x)[/latex] does not exist.
- We have [latex]f(5) = 3[/latex], [latex]\lim_{x \to 5^-} f(x) = 3[/latex], [latex]\lim_{x \to 5^+} f(x) = -2[/latex], and [latex]\lim_{x \to 5} f(x)= \text{DNE}[/latex].
- We have [latex]x = \infty[/latex]. What we are asking here is what happens to [latex]f(x)[/latex] as [latex]x[/latex] gets really big. In other words, what happens to the function as [latex]x[/latex] goes to infinity? Well, it looks like perhaps [latex]f(x)[/latex] is just heading towards zero. So we say [latex]\lim_{x \to \infty} f(x) = 0[/latex]. There is no such thing as a right hand or two-sided limit in this case, nor does it make sense to talk about [latex]f(\infty)[/latex]. So we just leave it as [latex]\lim_{x \to \infty} f(x) = 0[/latex]. This is the same thing as a horizontal asymptote.
- Similarly we are looking at what happens to [latex]f(x)[/latex] when [latex]x[/latex] goes more and more negative. It looks like maybe [latex]f(x)[/latex] is heading towards [latex]1[/latex], so we write [latex]\lim_{x \to -\infty} f(x) = 1[/latex].