Graphical limit example

We are asked what is a reasonable estimate for the limit of g of x as x approaches 3. So, what we have here in blue, this is the graph of y is equal to g of x, and we want to think about what is the limit as x approaches 3.

So, this is x equals 3 here. So, what you need to do is think about what is the limit of this function as we approach 3 from the left. And we're also going to think about what is the limit of this function, what does it appear to be from the graph as we approach this value from the right.

If it looks like we're approaching the same value, then that would be a reasonable estimate for the limit. The reason why they say reasonable estimate is because we're going to do it by inspection. We don't have a lot more information about the graph to know for sure, but from the graph, we can come up with a reasonable estimate. So pause this video and see if you can figure it out on your own.

All right, so let's think about it. Let's think about it in two parts. Let's think about approaching x equals three, or let's think about the limit as x approaches three from the left. As we approach x equals three from the left, our graph seems to... Our graph seems to, if I just eyeball it, seems to be approaching the value 4.

So from the left, it looks like we are approaching 4. Approaching, or let me write it this way: g of x approaching 4 as x approaches 3 from the left. Now, let's think about it from the right. As x approaches 3 from the right, what does it look like g of x is approaching? Well, it looks like g of x is approaching negative 3 as x approaches 3 from the right.

So I could say g of x approaching negative 3 as x approaches 3 from the right. And so we have two different values here. When we approach from the left, it looks like g of x is approaching 4, and when we approach from the right, it looks like g of x is approaching negative 3.

And so, because of that, we would say that this limit right over here... It's reasonable to say that this limit doesn't exist. So I'll write does not exist based on looking at this graph right over here. A good clue that it won't exist is the x value where we're trying to find the limit at. You see this jump in the graph right over here? It is discontinuous; it jumps from one value to another, which is a good clue that the limit might not exist here.