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# Limit at infinity of a difference of functions

## Video transcript

let's think about the limit of the square root of 100 plus X minus the square root of x as X approaches infinity and I encourage you to pause this video and try to figure this out on your own so I'm assuming you've had a go at it so first let's just try to think about it before we try to manipulate this algebraically in some way so what happens is X gets really really really really large as X approaches as X approaches infinity well even though this hundred is a reasonably a reasonably large number as X gets really large billion trillion trillion trillionth even larger than that trillion trillion trillion trillionth you can imagine that the hundred in this under the radical sign starts to matter a lot less as X approaches really really large numbers the square root of 100 plus X is going to be approximately the same thing as the square root of x so for for really really large large X's we can reason that the square root of a hundred plus X is going to be approximately equal to the square root of x and so in that reality we are going to really really really large X is in fact there's nothing larger where you can keep increasing X's that these two things are going to be roughly equal to each other so it would it's reasonable to believe that the limit as X approaches infinity here is going to be 0 you're subtracting this from something that is pretty similar to that but let's actually do some algebraic manipulation to feel better about that instead of that it's kind of hand wavy argument about the hundred not mattering as much when X gets really really really large and so let me rewrite this expression see if we can get manipulated in interesting ways so this is 100 plus X minus minus X so one thing that might jump out at you whenever you see one radical minus another radical like this is well maybe we can multiply by its conjugate and somehow get rid of the radicals or or at least transform the expression in some way that might be a little bit more useful when we try to find the limit as X approaches infinity so let's just see we can't just multiply it by anything arbitrary in order to not change the value of this expression we can only multiply it by one so let's multiply it by a form of one but a form of one that helps us that is essentially made up of its conjugate so let's multiply this let's multiply this x times the square root of 100 plus X plus the square root of x over the same thing square root of 100 plus X plus the square root of x now notice this of course is exactly equal to one and the reason why we like to multiply by conjugates is then we can take advantage of differences of squares so this is going to be equal to in our denominator we're just going to have we're just going to have the square root of 100 let me write it this way actually 100 plus X plus the square root of x and in our numerator we have the square root of 100 plus X minus the square root of x times this thing times square root of 100 plus X plus the square root of x now right over here we're essentially multiplying a plus B times a minus B will produce a difference of squares so this is going to be equal to this top part right over here is going to be equal to is going to be equal to this let me do this in a different color it's going to be equal to this thing squared minus - this thing - that thing squared so what's 100 plus x squared well that's just 100 plus X 100 plus X and then what's square root of x squared well that's just going to be X so minus X and we do see that this is starting to simplify nicely all of that over the square root of 100 plus X plus the square root of and these X's X minus X will just be nothing and so we are left with a hundred over the square root of 100 plus X plus the square root of x so we could rewrite the original limit as the limit the limit as X approaches infinity instead of this we've just algebraically manipulated it to be this so the limit as X approaches infinity of 100 over the square root of 100 plus X plus the square root of x and now it becomes much clearer we have a fixed numerator this numerator just stays at 100 but our denominator right over here it's just going to be it's just going to keep increasing it's going to be unbounded so if you're just increasing this denominator while you keep the numerator fixed you essentially have a fixed numerator with an ever increasing or a super larger an infinitely large denominator so that is going to approach that is going to approach 0 which is consistent with our original intuition