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# Limits at infinity of quotients with square roots (odd power)

AP.CALC:
LIM‑2 (EU)
,
LIM‑2.D (LO)
,
LIM‑2.D.3 (EK)
,
LIM‑2.D.4 (EK)
,
LIM‑2.D.5 (EK)

## Video transcript

let's say that f of X is equal to x over the square root of x squared plus 1 and I want to think about the limit of f of X the limit of f of X as X approaches positive infinity and the limit of f of X as X approaches negative infinity so let's think about what these are going to be well once again and I'm not doing this in an ultra rigorous way but more in an intuitive way is to think about what this function approximately equals as we get larger and larger and larger X's this is the case if we're getting very positive X is very in positive infinity direction or very negative is still the absolute value of those X's are very very very large as we approach positive infinity or negative infinity well the numerator we only have one term we have this X term but in the denominator we have two terms under the radical here and as X gets larger and larger and larger either in the positive or the negative direction this x squared term is going to really dominate this one you could imagine when X is a million you're going to have a million squared plus one the value of the denominator is going to be dictated by this x squared term so this is going to be approximately equal to x over the square root of x squared this term right over here the 1 isn't going to matter isn't going to matter so much when we get large very very very large X's and this right over here x over the square root of x squared or x over the principal root of x squared this is going to be equal to x over if I square something and then take the principal remember the principal root is the positive square root of something then I'm essentially taking the absolute value of x it's going to be equal to x over the absolute value of X for X approaches infinity or for X approaches negative infinity so another way to say this another way to restate these limits is as we approach infinity this limit we can restate it as the limit this is going to be equal to the limit as X approaches infinity of x over the absolute value of X now for positive X's the absolute value of X is just going to be X this is going to be X divided by X so this is just going to be one similarly right over here taking the limit as we go to negative infinity this is going to be the limit of x over the absolute value of X as X approaches negative infinity and remember the only reason why I was able to make this statement is that f of X and this thing right over here become very very similar or you can kind of say converge to each other as X approach as X gets very very very large or X gets very very very very negative now for negative values of X the absolute value of x is going to be positive x is obviously going to be negative and we're just going to get negative 1 and so using this we can actually try to graph we can actually try to graph our function so let's try to do that so let's say that is my y-axis this is my x-axis and we see that we have two horizontal asymptotes we have one horizontal asymptote at Y is equal to 1 so let's say this right over here is y is equal to 1 let me draw the let me draw that line as a dotted line we're going to approach this thing and then we have another horizontal asymptote at Y is equal to negative 1 so that might be right over there Y is equal Y is equal to negative 1 and if we want to plot at least one point we could think about what does f of 0 equal so f of 0 is going to be equal to 0 over the square root of 0 plus 1 or 0 squared plus 1 well that's all just going to be equal to 0 so we have this point right over here and we know that as X approaches infinity we're approaching this blue horizontal asymptote so it might look something like like this like let me do it a little bit differently let me do it a little bit there you go see this up so it might look something like this that's not the color I wanted to use so it might look something something like like that we get closer and closer to that asymptote as X gets larger and larger and then like this we get closer and closer to this asymptote as X approaches negative as X approaches negative I'm not drawing it so well so that right over there is y is equal to f of X and you can verify this by taking a calculator trying to plot more points or using some type of graphing calculator or something but anyway I just wanted to tackle another situation we're approaching infinity and or negative infinity and we're trying to determine the horizontal asymptotes and remember the key is just sit to say what terms dominate as X becomes very approaches positive infinity or negative infinity to say well what is that function going to approach and it's going to approach this horizontal asymptote in the positive direction and this one in the negative