1. Limits and Continuity
Introduction to Limits
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We just learned that if a function does not approach the same value from both sides, then the limit does not exist. But this is actually not the only scenario in which your limit won't exist. So here, I'm gonna show you all possible cases that you may encounter in which your limit does not exist. So let's go ahead and not waste any time here and jump right into these limits. Now the first limit that we're asked to find here is the limit of f of x as x approaches 3. Now this function might look familiar and as we approach 3 getting really, really close from that left side, my function is going to a y value of 1. Whereas from the right side, getting really, really close to 3, it's going to a y value of 4. Because these are not the same from either side, we know that this limit does not exist. Now this won't happen for every piecewise function, but it does happen commonly for piecewise functions that have a jump like we see here. We have a jump in between 1 piece transitioning to the next one. Now this is not the only case in which your limit won't exist, so let's go ahead and move on to our next limit here. Here we want to find the limit of f of x as x approaches 2. Now as we get really, really close to 2 from either side, my function appears to be going to infinity. Now if both sides are going to infinity as x approaches 2, wouldn't that just be my limit? Well, no. Here the limit does not exist because anytime that we have unbounded behavior, meaning that our function is going to infinity or negative infinity as x approaches c, our limit does not exist because infinity is not a number. So this happens commonly for rational functions that have an asymptote. And here we do have an asymptote right at x equals 2. So this is causing our unbounded behavior off to infinity, meaning that our limit does not exist. Now as you continue on in math, you may informally state the limit is equal to positive or negative infinity, but this is just used to describe the behavior of a function. And for our purposes here anytime you see unbounded behavior as x approaches c your limit does not exist. Now let's move on to finding our final limit here. Here we want to find the limit of f of x as x approaches 0. Now looking at our graph as we get really, really close to 0 from either side, our function appears to be doing something kind of crazy. It appears to be going up and down repeatedly. It's oscillating. Now something that you may be thinking here is couldn't I just zoom in really, really far to my graph to see exactly what it's doing as x approaches 0? And the answer is no because the further and further that we zoom in here, the more and more you'll see this function oscillating. So here, for the limit of this function as x approaches 0, the limit actually does not exist. And this will be true of any function that is oscillating near c like we see here. Now, this will happen often for functions that look exactly like this one, the sine of 1 over x or the cosine of 1 over x. So anytime you see a function like that, watch out for a limit that does not exist. Now that we know all the cases for which limits do not exist, let's continue practicing. Thanks for watching, and I'll see you in the next one.
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