Limits at infinity of quotients with square roots (even power) | AP Calculus AB | Khan Academy

Let's see if we can find the limit as x approaches negative infinity of the square root of four x to the fourth minus x over two x squared plus three. 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 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 zero 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 fourth here; that's a higher degree. But remember, it's under the radical here. So, if you want to look at it at a very high level, you're saying, okay, well, x to the fourth, but it's under, you're going to take the square root of this entire expression.

So, you can 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 the limit as x approaches negative infinity of...

So, let me just do a little bit of a side here. If I have 1 over x squared, let me write it, let me... 1 over x squared times the square root of 4 x to the fourth 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 fourth times the square root of 4 x to the fourth minus x.

And so this is equal to the square root of 4 x to the fourth minus x over x to the fourth, which is equal to, and all I did is I brought the radical in here. This is, you could view this as the square root of all this divided by the square root of 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, which is 1 over x to the third. So, this 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, 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.