 In this video, I want to talk about taking limits as x approaches positive or negative infinity when you have an exponential function. And to be able to calculate the in behavior, it's important to think of the graph of your typical exponential function. So if we have an exponential function whose base is a, which is greater than one, this will be a growth exponential function, it will be increasing along its domain. You'll see in that situation that if you take the limit as x approaches infinity, a to the x will equal infinity and it'll actually be growing very rapidly, much faster than any power function. But it's also important to know that as you go towards negative infinity, this thing actually will decay towards zero. A growth exponential function actually has a horizontal acetote at y equals zero. But only on the left side of its graph on the right hand side, it goes off towards infinity. Now, if you have a decay model, an exponential function whose base is less than one, you actually see the opposite behavior going on here. On the left hand side, it goes off towards positive infinity. On the right hand side, it goes on towards zero from above. But a decay model is just a reflection of a growth model across the y-axis. So if you know the growth model, then you'll know the decay model as well. So keep track of that. The key thing to remember here is if you ever see a to the negative infinity, that's going to equal zero. Again, assuming that a is greater than one. Or if you take like a to the infinity, that's going to equal infinity. Again, if a is greater than one, if a is like one-half, it's less than one. You get the reverse principle. So pay attention to that. So if we considered the limit as x approaches zero from the left of e to the one over x there, we would compute this thing as the limit as x approaches. Well, we really can just plug it in to the we can plug it into the rational expression because we can actually bring the exponential out. It's a continuous function. We get e to the limit as x approaches zero from the left of one over x. This is going to give you e to one over zero from the left. If you approach zero from the left of the exponential function here, excuse me, of the rational function one over x there, if you approach it from the left, that's going to be a negative infinity. So e to the negative infinity is going to equal zero, of course. Notice how this is different to if we take the limit as x approaches zero from the right of this thing, e to the one over x. In that situation, you pull out the exponential. You're going to get one over zero plus. That's going to be e to the infinity, and that then gives you infinity as well. So this is a function for which when you approach zero from the left, you're going to get zero. But if you approach zero from the right, you're going to get infinity. So this function has a one-sided vertical asymptote, which kind of makes sense because exponential functions have a one-sided in-behavior horizontal asymptote on the left side, but it explodes on the right hand side by composing an exponential with a rational function. You can force that behavior to happen with vertical asymptotes. It's a vertical asymptote on the right hand side, but it's a, it's not a vertical asymptote on the left hand side. Consider the limit as x goes to infinity of two e to the x over e to the x minus five. Now, the way that we can handle limits of these ratios is like we did before with rational functions. I could take the numerator and multiply it by one over e to the x, one over e to the x, and this would then end up giving you the limit. On the top, you'll get two on the bottom. You'll get, of course, one minus five times e to the x, like so, for which you could rewrite that using exponential rules. This becomes two over one minus five e to the negative x right here as x goes to infinity. If we plug in infinity in this situation, we get two over one minus five times e to the negative infinity. Like we saw a moment ago, if e is going to the negative infinite power, that's actually just a zero. You get two over one minus zero. That is two over one. The limit here turns out to be two. But personally, I like sort of a simpler approach. This mechanic works each and every time, but I like to think of it in terms of who's the fastest growing term here. In the numerator, the fastest growing term, well, there's only one, it's two e to the x. On the bottom, you get e to the x. That's the fastest growing term right there. And so as x goes towards infinity, nothing really matters in the end, except for, of course, the leading term. So you get two e to the x over e to the x, for which you see the same thing again, the e to the x cancel and the limit's going to turn out to be two. It's a much simpler calculation, but why is this method justifiable? It's because of all these details right here. It's kind of like proving the quadratic formula in an algebraic class. You compute, you get the quadratic formula by completing the square, but once you've done that, you don't have to do it over and over again. You could reuse the results of that quadratic formula. I should also mention that in this video, that is if we take x going towards negative infinity, if we had done that one, things will look a little bit different, of course, because if you plug in negative infinity in this situation, let's actually do that one. Let's take the limit as x approaches negative infinity of two e to the x over e to the x minus five here. If you plug in negative infinity, you'll get two times e to the negative infinity. You'll get e to the negative infinity minus five. e to the negative infinity is gonna be zero, so you get zero over zero minus five, and the end, this gives you zero as well. So that's a different result. It's important to remember with exponential functions that the behavior on the right is not the same as the end behavior on the left, and so therefore functions that have exponentials in them will likely inherit that property from its parent. All right, so let's look at one last example in this video here. Let's take the limit as x approaches infinity. We have e to the three x minus e to the negative three x over four times e to the three x plus e to the negative three x. In terms of dominance, I want you to be aware that as x goes to infinity, these terms are gonna go off towards zero because you're getting this e to the negative infinity there, and so this right here, because those terms are going off towards zero, they're recessive as x goes to infinity here, you're gonna end up with just the leading terms. The dominant term will be e to the three x over four e to the three x. The e's cancel out and we see the limit here is gonna equal one fourth. On the other hand though, if we take the limit as x approaches negative infinity, we see the exact opposite happening. Let me just copy this down right here, four e to the three x minus e to the negative three x, like so. So this time around, as x approaches negative infinity, these terms are gonna go off towards zero. The other terms aren't because if you plug in, if you plug in a negative infinity times a negative three, that's actually gonna be a positive power of e. So in terms of dominance, you actually end up with the limit, this time as x approaches negative infinity of negative e to the negative three x over negative e to the negative three x, for which we're gonna see that everything's gonna cancel out and this limit turns out to be a positive one, which is different than the one fourth we see there. So be very careful when you work with exponentials, look for these dominant terms, but also make sure you pay attention, am I going towards positive infinity or negative infinity, because the result is very different in all the direction you're going.