>> Here's a kind of a funny situation. It's the limit as X approaches 2 for this expression. Notice that if we use substitution we have 2 squared minus 4 in the numerator, and 2 minus 2 in the denominator, well that gives us the fraction zero over zero. Now this is called the zero over zero indeterminate form. And, and when this situation occurs a lot of times we have an opportunity for simplification, and we can actually find the limit. You see if, if, if X equals 2 causes the numerator and denominator to be zero, if, if, if 2 is a zero of an expression, then X minus 2 is a factor. So it might stand to reason that we have a factorization opportunity, and certainly we do. If we take that numerator and factor it we have X plus 2 times X minus 2, we bring down the denominator, and now we take advantage of the cancellation opportunity. So now we have the limit as X approaches 2 of X plus 2, and by substitution 2 plus 2 is 4 for that limit. Here's another similar situation. It's the limit this time as X approaches 1, and once again, sort of by observation, we can see if, if X is replaced with 1, 1 squared is 1, minus 3 time 1, that's negative 3. 1 minus 3 is negative 2, negative 2 plus 2, oh that's zero. And then in the denominator, 1 squared is 1, 1 minus 1, zero. Again we have the zero over zero indeterminate form, so we look for cancellation opportunities after factoring. And in the numerator we factor to X minus 1 times X minus 2. In the denominator the difference of squares factors to X minus 1 times X plus 1. And the X minus 1's cancel, so we have X minus 2 over X plus 1. And by substitution we're replacing X with 1, so we find that limit to be negative 1/2. The graphs of these two are rather interesting. When that, when these factors cancel it, it may make some sense that, that the, the functions are not defined at those values. And particularly on this first one when you construct the graph, it's a, it's actually a straight line on that first one. But the, the line has a hole in it at that place where it's undefined. And, and if you graph it in it's, you want to graph it down in it's original form, because that'll give you the, the hole in the graph at that particular point. Rather interesting situation. Here's another one though that's kind of related. The limit as X approaches 1 of this, and again, we would kind of go through and by substitution, substituting 1 for X in the numerator we would get this, which becomes 6. And in the denominator we get zero. So we have the 8 over zero indeterminate form. And in this situation we don't have that factorization opportunity. We don't have the target point being a factoring opportunity in that numerator, so we can't cancel, and therefore the limit does not exist in this circumstance.