In Section 1A, we saw how to go from a position graph to a velocity graph. Howev
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]. Nam