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• - [Voiceover] 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 2x 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 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 one over x squared and let's divide the denominator

• by one 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 wanna look at it at a very high level,

• you're saying, okay, well x to the fourth, but it's under,

• you're gonna 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,

• so this is going to be the limit,

• the 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 one over x squared,

• all right, let me write it.

• Let me just, one over x squared times the square root

• of 4x to the fourth minus x,

• like we have in the numerator here.

• This is equal to, this is the same thing

• as one over the square root of x to the fourth

• times the square root of 4x to the fourth minus x.

• And so this is equal to the square root

• of 4x to the fourth minus x

• over x to the fourth, which is equal to the square root of,

• and all I did is I brought the radical in here.

• 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 four minus,

• x over x to the fourth is one over x to the third.

• So this numerator is going to be,

• the numerator's going to be the square root

• of four minus one, 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 two.

• And then three divided by x squared is gonna be

• three 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 zero.

• One 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 zero.

• This over here is also going to be,

• this thing is also going to be approaching zero.

• We're dividing by larger and larger and larger values.

• And so what this is going to result in

• is the square root of four, the principal root of four,

• over two, which is the same thing

• as two over two,

• which is equal to one.

• And we are done.

- [Voiceover] Let's see if we can find the limit

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

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yukang920108 posted on 2022/07/05
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