![]() And so on our flowchart, we then continue to the right side of it and so here's a bunch of techniques for trying to tackle something Have negative one squared which is one minus two times negative one so plus two minus three which is equal to zero. You get negative one squared which is one minus negative one which is plus one minus two. Limit is x approaches negative one of this rational expression. What if when we evaluate the function, we get zero over zero? And here is an example of that. Most times, you do have a vertical asymptote there. So, it would be very, you will not have a vertical asymptote. Would be something like one over x minus x. Now, there are cases, very special cases, where you won't necessarily Might look something like this and then for values less than negative one or less than one I should say, you're gonna get negative values and so, your graph might look like something like that until you have this vertical asymptote. If x is greater than one, the denominator is going to be positive, and so, my graph and you would get this from What was going on there or even verify that it'sĪ vertical asymptote, well then you can try out some numbers, you can try to plot it, you can say, alright, I probably have a vertical asymptote here at x equals one. And at that point, if you wanted to just understand It says, okay, I'm throwing it, I'm falling into this So if we're talking about the limit as x approaches one of one over x minus one, if you just try toĮvaluate this expression at x equals one, you would get one over one minus one which is equal to one over zero. Where we just say the limit put that in a darker color. And what do we mean by vertical asymptote? Well, look at thisĮxample right over here. What happens if you evaluate it and you get some number divided by zero? Well, that case, you are probably dealing Plain vanilla functions that are continuous, if you evaluate at x equals a and you got a real number, that's probably going to be the limit. But in general, this is a pretty good rule of thumb. Or if you know visually around that point, there's some type of jump or some type of discontinuity, you've got to be a Some type of a function that has all sorts of special cases and it's piecewise defined as we've seen in previous other videos, I would be a little bit more skeptical. Just evaluate the function and it gives you a real number, you are probably done. Rational expressions like this or trigonometric expressions, and if you're able to You could just say, hey, can I just evaluate the function at that at that a over there? So in general, if you're dealing with pretty plain vanillaįunctions like an x squared or if you're dealing with Point right over here, the function is continuous, it's behaving somewhat normally, then this is a good thing to keep in mind. ![]() ![]() But if at that point you're trying to find the limit towards, as you approach this This will not necessarily be true if you're dealing with some function that has a point discontinuity like that or a jump discontinuity, or a function that looks like this. Of a continuous function which we talk about in previous videos, but sometimes, they aren't the same. ![]() The limit is a different thing than the value of the function. But then there's this little caveat here. And this flowchart says, if f of a is equal to a real number, it's saying we're done. So what this is telling us to do is, well the first thing, just try to substitute what So the goal is, hey, we want to find the limit of f of x as x approaches a. It looks a little bitĬomplicated at first, but hopefully it will make sense as we talk it through. What you see here is a flowchart developed by the team at Khan Academy, and I'm essentially going to To think about strategies for determining which technique to use. Videos and exercises we cover the various techniques for finding limits.
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