Video Transcript

Hi there, this is Ryan Malloy, here at the Worldwide Center of Mathematics. In this video, we're going to discuss one-sided limits in calculus. So, here we have F of X, which is a piece wise function. When X is less than two, it is defined by X squared. And when X is greater than or equal to two, it is defined by this weird looking function sine of X, to the power of X squared. Probably something you've never seen before. What we're being asked, is the one-sided limit, as X approaches two from the left of F of X. So, recall them, when we're dealing with limits, we're not entirely concerned with what actually happens at the function at that point. We only want to know what the behavior of the function is, as it approaches that value. So, with that in mind, consider that, if X is approaching two from the left. Then, within this limit, X will always be smaller than two. So, what that tells us, is that we can simply use this definition of F of X. And completely ignore this crazy sign of X function. So, this is equivalent to the limit as X approaches two from the left, of just X squared. Well, this is easy to compute. We can simply plug in two for X squared, and see what we get. Two squared equals four, easy enough. Now, the other situation, in which one sided limits are useful, is when we have a function of X in the denominator. So, here we have G of X, is equal to one over one minus X. At the point X equals one, we have an infinite discontinuity. What we want to know, is if we approach one from the right side, or from the positive side. Is the function going to positive infinity, or to negative infinity? We could simply graph this on a calculator, but assume you're able to do that. We should be able to find a rigorous mathematical way, determining if the function goes to positive or negative infinity. So, let's rewrite this, we've got one minus one plus. This may seem like a bit of a weird notation, but let's just see where we're going. One plus, what does that mean? It means some value that is infinitely close to one, but slightly larger than that. It might be helpful to think this is being 1.00001, or something like that. But what this tells us, is we have one in the numerator. And when we have one minus some value that is slightly larger than one. What we get is something that is very close to zero, but slightly smaller than that. So, it's a very slightly negative number. This is one way to think about it. So, this is essentially equivalent to negative one over zero. Which tells us that this function goes to negative infinity, as we approach one from the right. My name is Ryan Malloy, and we've just discussed one-sided limits in calculus.