PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
A greatest integer function is the greatest integer less than or equal to a given function. This function can be found on a calculator to find values of f(x) that are all integers. The basic strategy of finding limits by calculating values of f(x) remains the same. In order to find the limit of a function as x approaches a point p, plug in values of x into f(x) as x approaches a from both the left and right (when x is less than but getting closer and closer to p, and when x is greater than but getting closer and closer to p). The limit exists if f(x) approaches the same number when x approaches p from the left and from the right (i.e. when the left-hand and right-hand limits are the same). Remember, limits are about what happens when a function gets close to a certain point, but this can be different from what actually happens at that point. For example, if the limit of f(x) as x approaches 5 from the left is 7, and the limit of f(x) as x approaches 5 from the right is also 7, f(5) could possibly be a number other than 7, like 32, but the limit of f(x) at 5 would be 7 because the left-hand and right-hand limit equals 7.
Let’s do a harder example. I want to use a table to find a couple of limits. First as x approaches 8 from the left of h(x), when h(x) is the greatest integer less than or equal to 100 over x² plus 66. How do you enter this greatest integer function on your calculator?
Your calculator has a function called int, and that will give you the greatest integer less than or equal to that number. So for example the greatest integer greater than or equal to 2.3 is 2. So int of 100 divided by, you’re going to need to get your x² minus 16x plus 66, and that will do it for you.
Let’s create a table about our us and see what the result is. Here we are on the TI 84 again. Let me enter this function, I have to hit the y equals button first. I’m going to enter right here. Where do I find int? It’s going to be in the Math menu, under numbers. Go to the right. It's number 5. So I hit 5 and I have int of 100 divided by (x² minus 16x plus 66). Let’s look at the table.
We can actually see a little piece of the graph. It's kind of cool. We want x to approach 8 from the left, so let’s see what happens. Let me try something; let’s just see the way this function behaves in general. I plug in 5 and I get 9. 6, I get 16, 7, I get 33. What about 7.5? 44.
This is a very strange function. 7.9, 49. 7.99, 49. 7.999, 49 again. What you can see is happening is, it’s kind of levelled off. It does keep increasing as you increase towards 7. Once you get into the 7.9, 7.99, 7.999, and I’ll go back up here, and I’ll do 7.9999, just to make sure. It’s still 49. Let's record these results on the board. I’ve written the results on the table.
Remember we’re letting x approach 8 from the left, so I have numbers 7, 7 ½, 7.9, 7.99, 7.999, these are approaching 8. And look at what my h(x) values are doing. I have 33, 44, and then we notice that once we get up to 7.9, 7.99, 7.999, we stayed at the value 49. So it seems like this limit is going to be 49. We’ll write that down. Let’s see what happens as we approach 8 from the other side; from the positive side.
We’ll go back to our TI 84. Okay we're back in our TI 84, I want to approach 8 from the positive side, so let me start with a value like 9. I get 33. So 8.5 is a little closer, 44. 8.1, a little closer still, I get 49. 8.01, 49. It looks like it’s doing the same thing form the right as it was doing from the left. Let me try 8.001.
I’m going to conclude that it does the same thing from the right as it does from the left. That it’s actually approaching the value of 49. But I want you to see something very interesting. If I plug exactly 8 in, 50. Isn’t that interesting? It actually has a different value at 8 than the value that it’s approaching as x approaches 8.
Let’s write these results on the board and then we’ll wrap it up. I’ve got my values on the board. I have x approaching 8 from the right. Now what do the h(x) values do. Well just like before, 33, 44, 49, 49, 49, these values are approaching 49. And no matter how many zeros we put, as long as we put a 1 after that, we’re still going to get the value of 49. So I would say that the limit as x approaches 8 from the right of h(x) is 49.
That means that the two sided limit exists. We notice that as x approaches 8 from the left we got 49, as x approaches 8 from the right we got 49. So the two sided limit is going to be 49. What’s interesting about this function is that h(8), the value that the function actually has at 8, is 50.
The thing you have to remember about when you’re calculating limits, is what actually happens at the number, in this case, 8. What actually happens at 8 doesn’t matter. You have to focus on what’s happening as you get close to 8. What’s happening at 8 may be different. So remember, as x approaches 8 we get 49.