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# Limits at infinity of quotients with square roots (even 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 see if we can find the limit as X approaches negative infinity of the square root of 4x to the fourth minus x over 2 x squared plus 3 and like always pause this video and see if you can figure it out well whenever we're trying to find limits at either positive or negative infinity of rational expressions like this it's useful to look at well what is the highest degree term in the numerator or in the denominator or actually in in the numerator and the denominator and then divide the numerator and the denominator by that highest degree by X to that degree because if we do that then we're going to end up with some constants and some other things that will go to 0 as we approach positive or negative infinity and we should be able to find this limit so what I'm talking about let's divide the numerator by 1 over x squared and let's divide the denominator by 1 over x squared now you might be saying wait wait I see an X to the 4th here that's a higher degree but remember it's under the radical here so if you want to look at it a very high level you're saying okay well let X to the 4 it's under its you're going to take the square root of this entire expression so you could really view this as a second degree term so the highest degree is really second-degree so let's divide the numerator and the denominator by x squared and if we do that dividing so this is going to be the same thing as so this is going to be the limit the limit as limit as X approaches negative infinity of so let me just do a little bit of a side here so if I have if I have 1 over x squared all right let me write it let me 1 over x squared times the square root of 4x to the 4th minus X like we have in the numerator here this is equal to this is the same thing as 1 over the square root of x to the 4th times the square root of 4x to the 4th minus X and so this is equal to the square root of 4x to the 4th minus X over X to the 4th which is equal to the square root of and all I did is I brought the radical in here this is you could view this as a square root of all this divided by the squared this which is equal to just using our exponent rules the square root of 4x to the fourth minus X over X to the fourth and then this is the same thing as 4 minus x over X to the fourth is 1 over X to the third so this numerator is going to be the numerator is going to be the square root of 4 minus 1 X to the third power and then the denominator is going to be equal to well you divide 2x squared by x squared you're just going to be left with 2 and then 3 divided by x squared is going to be 3 over x squared now let's think about the limit as we approach negative infinity as we approach negative infinity this is going to approach 0 1 divided by things that are becoming more and more and more and more and more negative their magnitude is getting larger so this is going to approach 0 this over here is also going to be is this thing is also going to be approaching 0 we're dividing by larger and larger and larger values and so what this is going to result in is the square root of 4 the principal root of 4 over 2 which is the same thing as 2 over 2 which is equal to 1 and we are done