Calculus Crash Course: Part 2, Limits
Much about Calculus involves examining the way functions behave over the course of some interval or at some point. One way to evaluate a function at a particular point is to just plug in a value. Consider the following function...
f(x) = 3x - 2
Suppose we wanted to know what f outputs when we use 7 as an input. By plugging in 7 for x, we obviously get a value of 19 for f(7). Easy stuff.
Another method is using limits. What I am about to say may seem completely unnecessary and a roundabout way of doing things. But stick with me on this...
If we want to know what the value of f is at x=7, then we can actually figure that out by using the plug-in-numbers method for values near 7. The smaller the difference between the number we pick and 7, the closer our output will be to the output of f(7).
Consider f(x) = 3x - 2 again...
| x | f(x) |
| 8 | 22 |
| 7.5 | 20.5 |
| 7.25 | 19.75 |
| 7.125 | 19.375 |
| 7.0625 | 19.1875 |
| 7.03125 | 19.09375 |
| 7.015625 | 19.046875 |
| ... | .... |
The closer we get to x=7, the closer we get to f(x) = 19. Because we're starting the values of x a little greater than 7 and slowly making them smaller toward 7, we call this "the limit from the right".
We can do the same thing, but starting smaller than 7 and increase them towards 7...
| x | f(x) |
| 6 | 16 |
| 6.5 | 17.5 |
| 6.75 | 18.25 |
| 6.875 | 18.625 |
| 6.9375 | 18.8125 |
| 6.96875 | 18.90625 |
| 6.984375 | 18.953125 |
| ... | .... |
Because we're starting x smaller than the target value and increasing it towards the target value, we call this "the limit from the left".
We write limit like this...
To notate limits from the left and right individually, we use a small + or - next to the limit value...
Eventually the difference between x and 7 will become infitesimally small and f(x) will be infitesimally close to 19 from either direction.
However, not all functions give the same output when the input approaches from the left and from the right. Here is one classic example: evaluate the following function at x = 0...
f(x) = 1 / x
Well, for one, you can't simply plug in 0 as input. 1 / 0 is undefined. But what does undefined mean? Why isn't it just infinity? Let's look at the limit of 1 / x as x approaches 0 from the right...
| x | f(x) |
| 1 | 1 |
| 0.1 | 10 |
| 0.01 | 100 |
| 0.001 | 1000 |
| 0.0001 | 10000 |
| 0.00001 | 100000 |
| 0.000001 | 1000000 |
| ... | .... |
It grows without bound. In this case...
However, when we approach from the left, something else happens...
| x | f(x) |
| -1 | -1 |
| -0.1 | -10 |
| -0.01 | -100 |
| -0.001 | -1000 |
| -0.0001 | -10000 |
| -0.00001 | -100000 |
| -0.000001 | -1000000 |
| ... | .... |
It grows without bound again, but negatively. So therefore the limit from the left is...
Infinity and negative infinity are most certainly not the same numbers. If you get two totally different limits when you evaluate the limit from the left and from the right, then there is no defined limit, such as in this case (hence the "undefined" result of anything divided by 0 that you've always accepted as gospel since elementary school). Now you really know how to evaluate a number divided by 0.
So in most cases, plugging in a number to figure out the value of a function will suffice, but there are cases where it will not. That is when limits may be of use to you. Limits also open up an entirely new door called derivatives that we will learn in the next section of this tutorial.
