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AP®︎/College Calculus AB
Course: AP®︎/College Calculus AB > Unit 1
Lesson 15: Connecting limits at infinity and horizontal asymptotes- Introduction to limits at infinity
- Functions with same limit at infinity
- Limits at infinity: graphical
- Limits at infinity of quotients (Part 1)
- Limits at infinity of quotients (Part 2)
- Limits at infinity of quotients
- Limits at infinity of quotients with square roots (odd power)
- Limits at infinity of quotients with square roots (even power)
- Limits at infinity of quotients with square roots
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Limits at infinity of quotients with square roots (odd power)
Sal finds the limits at positive and negative infinity of x/√(x²+1). Since the leading term is raised to an odd power (1), the limits at positive and negative infinity are different. Created by Sal Khan.
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at1:39, why do we take the principal square root of x and not just the square root of x?(49 votes)
The definition of a function is that an input has one output.
So, if f(x)=sqrt(x), unless we used the principal square root, f(4)= 2 and -2
If this is a function, the input 4 cannot have two outputs!
That is why when using the square root in a function, we use the principal square root.(98 votes)
Hey, the limit when x-> negative infinite shouldn't be just 1? Since it will be negative infinite divided by negative infinite.(8 votes)
It's not negative infinity in the denominator because of the absolute value, which comes from taking the principal root of the any number squared.(13 votes)
At2:00, Sal says that principal root means that we're taking the positive answer.
Is it that we're ignoring the negative, or is it that it doesn't exist (i think it's the first one- but id dont know.) Also if we're ignoring the negative answer, then what will the standard equation look like?
Finally, if we just take the square root, wouldn't it be easier, because then the answer to "limiting x to infinity and negative infinity" will be the same(6 votes)- Square roots of real numbers are always assumed to be positive. The real number line has a very special property - in analysis it is called being an "ordered field". The ordering means that saying one number is "less than" another, a < b, or "greater than, a > b, makes sense. The complex numbers are not ordered. We can talk about their lengths being > or < but saying "w < z" doesn't make sense. As such (and from convention), we always define the square root to be a positive value, and use the minus/negative sign to explicitly state that the answer is negative. So \sqrt(9) is always 3. If we want -3, we must explicitly state " - \sqrt(9)" to get - 3. As the counterexample, \sqrt(- 5 + 12i) does have meaning, as both 2 + 3i and -2 - 3i = - (2 + 3i) square to get - 5 + 12i. However, since the complex numbers are not ordered, when we talk about the square root or use that symbol, we must recall that both 2 + 3i and its negative are implied. I think Sal is talking about the principal root to remind you that in general, when you square things or take square roots, you need to be aware of the quantity and quality (sign) of your solutions and make sure the notation reflects what you really want to say.
For the second question, standard notation means that the symbol " \sqrt( x^2 + 1) " returns a positive answer. If you want a negative value, you must explicitly state " - \sqrt(x^2 + 1) ". The reason he uses | x | instead of just x is this coupled with order of operations. When you take any x value and square it (and add one), the answer is always positive, which should make sense because you can't take the square root of a negative number. But when we take the square root, it is assumed the resulting value is positive, so we use the absolute value sign to reflect that. If it helps, note that if we allowed both the positive and the negative, the answer would not be x/x, but rather x / ( +/- x) which is no longer a function (one input = two outputs), but instead the implicit graph of two lines.
I hope this helps... underneath your questions are some very deep concepts about the real numbers and our common definition of various symbols.(16 votes)
How did he conclude that the square root of x was positive? xD(7 votes)- At1:38, It was the principal square root of x^2 and not just the square root of x.(6 votes)
How do you calculate the function of a oblique/slant asymptote?(3 votes)
Not to be disrespectful to Matthew, but I think he made mistakes from ex. (3) (x^3)/(x+2) and on. He consistently left out the constant term of slant asymptotes. For example, he said that the asymptote was y=x^2-2x. But I found it out to be y = x^2 -2x +4.
I did a quick google search on how to find slant asymptote, and I found a simpler method.
http://www.purplemath.com/modules/asymtote3.htm
http://www.purplemath.com/modules/asymnote.htm
Basically you just do a long division, and the asymptote eqn is just the polynomial part of the answer without the remainder, which makes sense bc the remainder is divided by infinity > becomes 0.(2 votes)
- Can someone tell me what an asymptote is?
I don't really get what it is(4 votes)
It is a line, that comes closer and closer to a function-graph as you move to (plus or minus) infinity, but does not touch or cross it.
https://upload.wikimedia.org/wikipedia/commons/c/c4/1-over-x-plus-x.svg(4 votes)
The square root of x^2 is +-x. So why did we take the abs value of x? Shouldn't it be +-x(0 votes)- Also note that we're doing f(x). Functions do not allow ± because we'll end up with 2 y-values for most x-values. Functions give only 0 or 1 y-value per x-value. To avoid any controversy, we simply stick to the "positive square root" (Sal even uses this term in the video).(3 votes)
1:37
If you’re taking the principle root or square root or whatever of √ x², why is it |x| instead of just positive x? Like if I had 2² and then I took the square root of that, wouldn’t the square and the square root just cancel each other out and the answer would be two?(1 vote)
Suppose we define √x² to be x. This means that whatever I substitute as x, my output should also be x. Now, if I substitute x = 2, I get √(2)² = 2. Works out. However, if I substitute x = -2, I get √(-2)² = 2. Clearly, this is a problem. So, see that for x = 2 or -2, the output is 2. Hence, √x² comes out to |x| and not just x.(6 votes)
- How do you determine what the right-end and left-end behavior models of a function will be?(1 vote)
How de we know when a function would have a horizontal asymptotic, a vertical asymptotic or both?(2 votes)- A horizontal asymptote occurs when the limit of the function (the y-value) approaches a constant as x -> infinity (or -infinity). This will happen for rational functions (a polynomial divided by another polynomial) if the degree of the denominator is greater than or equal to the degree of the numerator. It can also occur with special functions like e^x (horizontal asymptote of y = 0 as x -> -infinity).
A vertical asymptote occurs when the function goes to infinity as the x-value approaches some finite number. Usually, this is because there is a fraction where the denominator becomes zero but the numerator does not. For example, f(x) = (x + 2)/(x - 3) has a vertical asymptote of x = 3, since the numerator wants to be 5 and the denominator wants to be 0 at x = 3. Again, there are also some special functions with vertical asymptotes (like ln(x) at x = 0), that you should probably just memorize.(3 votes)
1:39
Video transcript
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.