Calculus: The Limit Of A Sequence – With Square Roots

Hello my lovelies, it's Susanna and today I want to show you how to find the limit of this sequence here where we have to deal with square roots. We want to find the limit as n approaches infinity of this expression here. And at the beginning I would always just check first what happens with my expression if my n gets bigger and bigger. Maybe we can find the limit immediately. Maybe not. Probably not. But let's see what happens. Okay, so n is getting bigger and bigger. So my n in here is getting bigger and bigger and will go up to infinity. If I square something that is getting bigger and bigger, this whole thing is going to be even bigger. If I add one, it doesn't change a whole lot. So this thing in here is still going up to infinity. And if I take the square root of something that is super big, that doesn't change anything. It's going to be super big anyway. So all of this is going to up to infinity. Then I have my minus. What about my second square root here? My n is getting bigger and bigger. So it goes up to infinity. Same here. If I square it, it's still going to be super big. If I subtract one, it doesn't change a whole lot. I still have a super big number in here. I take the square root, it stays super big. So all of this square root is going up to infinity as well. So in total I have infinity minus infinity. And some of you might say, yeah, this is probably going to be zero. But we don't know. This could be anything. If you have infinity minus infinity, it could be zero. The result, yes. But it could also be infinity itself if this expression grows faster. It could be negative infinity if this expression grows faster. It could be any number. It could be two. It could be one over two. We just don't know. So every time you end up in a case where you have infinity minus infinity, you don't know anything. So we cannot find the limit right now as our expression looks like here. We have to change it, rewrite it, find a different version of this expression so that we can work with it and find the limit. How can we do that? Well, there is a trick that we can use. Every time we have this structure here where we have a square root minus another square root, or it also works if you have a square root plus another square root, you can do the following. You can multiply this by a big fraction. And this might look weird if you see it for the very first time. But trust me, the more and more you see of these problems, one day you would be like, oh, I have a square root minus another square root. I know exactly what to do. So we're going to multiply by a fraction. And the question is, what are we going to write down on this fraction? Now, we're going to take our expression here almost exactly as it is here. But we're going to change one little thing. We take the first square root, write it down exactly as it is, we don't change anything here. But now, instead of the minus, we're going to write down a plus, we will see how this helps us, it definitely will. But we just change this symbol. And also if you had a plus here, you're going to change it into a minus then. So just change this, but keep the rest. So don't change anything in your square roots. Okay, this is the numerator. And now we take exactly this thing and write it down in the denominator as well. We have to have the same things here in the numerator and in our denominator. How does this help us now? I mean, it looks way more complicated than before. But trust me, we will be able to find our limit as n approaches infinity. We just have to multiply this thing by this fraction now. And we do it by taking this thing, we can write it in parentheses, we're going to multiply by the numerator, so I can also write it in parentheses. And the denominator will stay the same. So our result is going to be a big fraction, we can take the denominator as it looks this beautiful thing and write it down here exactly as it was. And now we're going to multiply this by this. But it's easier than you might think. Because if we compare these two, we have the same things in here. But one is connected by a minus, and one is connected by a plus. Do you know this structure? It is part of the formula of the difference of squares. If you have two parentheses that you multiply by each other, you have a minus b in one of the parentheses, you have a plus b in one of the parentheses, then the result is going to be a squared minus b squared, the difference of squares. So in our case, we want to multiply something, we have the same things in the parentheses, but one here we have a minus, and the other one we have a plus, we know the result is going to be the first thing squared minus the second thing squared, which is great, because we work with square roots. And if we square these things, we're not going to have any square roots anymore. So we take the first thing and square it, which means that we get rid of the square root and only n squared plus one, so everything that is in our square root is what is going to be left here. Then we have a minus, okay, and then we square the second thing. So the second square root, if we square it, we can get rid of the square root, and only what is in here is left. But be careful, because of the minus, I'm going to write down the result in parentheses here. So I take my n squared minus one here. Okay, no more square roots here. We can simplify this expression a little bit, because before we keep on going, we have n squared here plus one, then minus n squared, if we get rid of the parentheses, and be careful, minus minus one is going to be plus one. So in total, in our numerator, we have n squared minus n squared, and one plus one equals two. So we just have two in the numerator. And of course, still this beautiful denominator, I'm not going to write it down here, because we can see it here already. Okay, so now we have a different version of our expression, we just have one big fraction, we only have the two in here, which is great. And now let's try again to find the limit as n approaches infinity. n is getting bigger and bigger. What does my two do if n is getting bigger and bigger? It doesn't care at all. The two, there is no n in here, it doesn't change. So the two just stays at the two, there is no n that could affect, have any effect on here. It looks different in the denominator, there are some n's, so my n is getting bigger and bigger, it will go up to infinity. If I square it, we know it's going to get bigger and bigger, plus one as well. The square root doesn't change anything, so my first square root is getting bigger and bigger, and will go up to infinity. This time I have a plus here. Okay, my second square root, same thing. n will go up to infinity, everything will get bigger and bigger, my square root in total will go up to infinity, but this time, as I said, we have infinity plus infinity, which is definitely going to get bigger and bigger, and will go up to infinity. So in total, we have two in the numerator, we have infinity in the denominator, which means that the two just stays the same, it doesn't change at all, we always have two, and divide this by a number that is getting bigger and bigger, which means that this fraction in total is going to get smaller and smaller, and will approach zero then, and this is our limit of our sequence. I hope it helps you. If you have any questions, please let me know in the comments. I wish you a wonderful day, and I hope to see you in one of my next videos. Take care!