 ## Subtitles section Play video

• Let's say that f(x) is equal to x over the square root of x^2+1

• and I want to think about the limit of f(x) as x approaches positive infinity

• and the limit of f(x) as x approaches neagative 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'es.

• This is the case if we're getting very positive x'es, in very positive infinity direction or very negative.

• Still the absolute value of those x'es are very very very large as we approach positive infinity

• or negative infinity.

• Well, in the numerator we only have only 1 term -- we have this x term --

• -- but in the denominator we have 2 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 can imagine, when x is 1 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 so much when we get very very very large x'es.

• And this right over here -- x over the square root of x squared or x over the principle root of x squared --

• -- this is going to be equal to x over --

• -- if I square something and then take the principle root --

• -- remember that the principle root is the positive square root of something --

• -- then I'm essentially taking the absolute value of x.

• This is 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'es 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 1.

• Similarly, right over here, we take 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.

• Remember, the only reason I was able to make this statement is that f(x) and this thing right over here

• become very very similar, you can kind of say converge to each other,

• 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 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 2 horizontal asymptotes.

• We have 1 horizontal asymptote at y=1,

• so let's say this right over here is y=1,

• let me draw that line as dotted line,

• we're going to approach this thing,

• and then we have another horizontal asymptote at y=-1.

• So that might be right over there, y=-1.

• And if we want to plot at least 1 point we can think about what does f(0) equal.

• So, f(0) is going to be equal to 0 over the square root of 0+1, or 0 squared plus 1.

• Well that's all just going to be equal to zero.

• 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 this.

• Let me do it a little bit differently. There you go.

• I'll clean this up. So it might look something like this.

• That's not the color I wanted to use.

• So it might look something 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 infinity.

• I'm not drawing it so well.

• So that right over there is y=f(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 to say what terms dominate

• as x 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.

Let's say that f(x) is equal to x over the square root of x^2+1

Subtitles and vocabulary

Click the word to look it up Click the word to find further inforamtion about it

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

• 2 1
yukang920108 posted on 2022/07/05
Video vocabulary