Sunday, June 3, 2012

One Sided Limits

Continuing on from my previous post in my series about limits in calculus, in this one I would like to better explain the concept of one-sided limits.  In my last post, I introduced you to the correct notation for writing a limit, though I mentioned that the notation that I was describing did not include any description of direction.

Recall my explanation for a limit, where I basically said that a limit (L) is the value that a graph appears to have at a specific point, assuming you were to follow along the curve and approach that point.  Kind of like making a prediction of the value of a point, based on what you know the curve is doing around it.  Now, with that definition in mind, you must realize that you can obviously approach the point in question from either side on the curve.  For simple equations (or more generally, continuous ones), you will approach the same limit value as you approach the "a" value from either side.  And this should intuitively make sense.  However, what I want you to consider now is something like the following example (a piecewise function):

What you can see in this example is that the value of f(x) for all values 5 and less equals 2, and for all values greater than 5 is 4.  Also, note the open and closed circle notation of the two lines.  The closed circle indicates that it includes 5 on that line.  The open circle indicates that it does not include 5 on the line, yet it contains all values up to 5, getting infinitely closer to it but yet never quite touching it.

So, from this example, now let me put the question to you: what is the limit of f(x) as x approaches 5?

To solve this question, you must consider one-sided limits, and examine the graph from either side of x=5.  If you consider what the graph "looks like it's going to do" as it approaches 5, you see that it kind of looks like it might be doing two different things.  (In fact, I will discuss continuity in separate post, though what we will see is that if you examine the limit at a value "a" from either side, and you determine that it is the same limit value for each one sided limit, then that is proof that the curve is continuous at "a").  If you approach x=5 from the left, you can see that it looks like the line will continue on as it has been doing, and the limit of f(x) when x=5 will be 2.  This is correct.  Consider the other part of the function, and approach x=5 from the right.  What does the limit appear to be?  Well, similarly to the left part, the right part appears to keep doing what it's doing, and the limit looks like it will be 4.  This is also correct.

So then, what I've shown here is that if I ask you to find the limit of f(x) as x approaches 5, you cannot give a single answer.  There are obviously two seemingly correct answers to this.  And so to distinguish between the two, you use the concept of one sided limits.  (Again, as I will explain in my post on continuity, you can obviously see by looking at the graph that the one sided limits on each side of 5 are not the same value, and therefore you can say that the graph is discontinuous at x=5.  Looking at the graph, you can immediately see this is true.)  You would say "as x approaches 5 from the right", or "as x approaches 5 from the left."  It is not overly complicated, but it is something to be aware of.

To make things less confusing, of course there is a correct notation to indicate one sided limits.  Thankfully, it is only a very small modification to the limits notation that I have already presented to you.  Specifically, all you have to do is add a "+" or a "-" to indicate which side of the point "a" you are approaching from.  It may be more helpful to consider the notation as referring to "all values greater than a" for the +, and "all values less than a" for the -.

Here is what the one sided limit notation looks like, for when x approaches a from either the right or the left, respectively:


This is a commonly used notation in calculus, and you will frequently see it when you are doing your limits mathematics homework.  Hopefully, now you will be able to understand the notation when you see it and be able to instantly translate it and apply it to what you see in the graph.  As I mentioned above, I will next provide a more formal discussion about the concept of continuity of graphs, and how you can determine whether a graph is continuous or not based on its equation.  As always, please remember to +1 this post if you found it useful!  Thanks!

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