JAMES SOUSA: We want to use the provided graph of f of x to determine the function value f of 7, then determine the following four limits. To determine the value of f of 7 or evaluate f of 7, we want to determine the function output when the input or x value is positive 7. So looking at the graph, notice here's where x equals 7. Notice how we have a hole in the graph here, and we have a solid point on the graph here. 

Therefore, when x equals 7 or the input is 7, the function value is not here where the hole is. It's up here, where the point is closed. And therefore, f of 7 is equal to the function value or y value of 7. Next, we have a one-sided limit. We have the limit as x approaches 7 from the positive side or right side of f of x. 

So we already know this is where x equals 7. But to determine this limit, we're only concerned about approaching 7 from the positive side or right side, which would be from this direction here. Notice, from this direction, as we get closer and closer to x equals 7, the function value or y value is approaching positive 7. And therefore, this one-sided limit is equal to 7. 

Next, we have the limit as x approaches 7 from the negative side or left side of f of x. So once again, we're still approaching x equals 7, but now only from the negative side or left side, which would be from this direction here. Notice from this direction, as we get closer and closer to positive 7, the y value or function value approaches 2. And therefore, this limit is equal to 2. 

And next, we have the limit as x approaches 7 of f of x. Notice for this limit to exist, we'd have to be approaching the same function value or y value from the left and right of positive 7. We can see graphically that we're not approaching the same function value. From the right, we're approaching positive 7. And from the left, we're approaching positive 2, which we already determined by looking at these one-sided limits. So because these two one-sided limits are not the same, the limit as x approaches 7 of f of x does not exist. 

And now for the last example, we have the limit as x approaches 8 of f of x. So going back to the graph one more time, notice here's where x equals 8. And paying close attention to the notation, we have to approach 8 from both the right side and the left side. And we have to be approaching the same function value for this limit to exist. 

So as we approach from the right and as we approach from the left, notice how we are approaching the same function value. It would be this function value here where y equals 5. And therefore, the limit as x approaches 8 of f of x equals 5. I hope you found this helpful.