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Limits: A Graphical Approach - Problem 1

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

In order to find the limits from a graph of a function, you simply observe what happens to the graph as it approaches a certain point. By following the curve of the graph, you can see what f(x) gets closer to as x approaches that point. To find the left-hand limit, simply observe what happens as x approaches the point from the left-hand side of the graph, where x is less than that point, and to find the right hand limit, observe what happens as x approaches the point from the right-hand side of the graph, where x is greater than that point. Recall that the limit of a function as x approaches a point is not necessarily the same as the value of a function at that point. When looking at limits, you simply look at what value f(x) is getting closer and closer to.

It’s pretty easy to find limits if you have the graph of a function. Here I’ve got the graph of a function called f(x) and I’m asked to determine 3 limits and then the value of the function at zero. First of all, notice that these two limits I’m asked for in a and b on the left hand and the right hand limit as x approaches zero. So we’re focusing here on zero. As x approaches zero what happens?

As x approaches zero from the left, we’re on this part of the graph. So as x approaches zero, we come down and then up again and we approach this point here. As x approaches zero from the left, the values are approaching 2. What about from the right? Well from the right, we’re on this piece of the function, so as x approaches zero from the right we’re going down this branch here and approaching -2.

Now you learned previously that the only way that the two sided limit, the limit as x approaches zero of f(x) exists is if these two limits exist; the one sided limits exist and if they're equal. But they're not equal. So this limit does not exist.

Finally let’s observe that f is actually defined at zero. This closed circle means that f(0) is -2 and so that’s important. It’s important to know that when you’re calculating limits, what actually happens at zero, is not important. It’s what happens as x approaches zero that’s important. So keep these ideas separate in your mind. What happens at zero and what happens as x approaches zero.

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