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Limits at infinity of quotients with square roots (odd 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)
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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  • orange juice squid orange style avatar for user Richik Bandyopadhyay
    at , why do we take the principal square root of x and not just the square root of x?
    (46 votes)
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    • piceratops ultimate style avatar for user Mathuss
      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.
      (91 votes)
  • leafers tree style avatar for user Luan Cristian Thums
    Hey, the limit when x-> negative infinite shouldn't be just 1? Since it will be negative infinite divided by negative infinite.
    (7 votes)
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  • leaf yellow style avatar for user Zorawar Singh
    At , 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)
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    • blobby green style avatar for user Andrea
      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.
      (14 votes)
  • male robot hal style avatar for user Allen Yun
    How did he conclude that the square root of x was positive? xD
    (7 votes)
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  • mr pants teal style avatar for user Sven
    How do you calculate the function of a oblique/slant asymptote?
    (3 votes)
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  • piceratops ultimate style avatar for user Darren Yao
    Can someone tell me what an asymptote is?
    I don't really get what it is
    (4 votes)
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  • mr pink red style avatar for user Arya Das
    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)
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    • mr pants teal style avatar for user Alexander Lee
      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)
  • blobby green style avatar for user David Wolfzorn
    How do you determine what the right-end and left-end behavior models of a function will be?
    (1 vote)
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  • starky ultimate style avatar for user Ashu Bhadouria
    How de we know when a function would have a horizontal asymptotic, a vertical asymptotic or both?
    (2 votes)
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    • blobby green style avatar for user robshowsides
      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)
  • old spice man green style avatar for user Karandeep Singh Dhillon
    You showed us
    How to solve f(x)=x/Principal square root of x^2+1
    But
    How to solve f(x)=x/Principal square root of x^3+1
    and
    How to solve f(x)=x/Principal square root of -x^3+1
    (1 vote)
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    • leaf green style avatar for user ArDeeJ
      So we have
      f(x) = x / √(x^3 + 1)
      = √(x^2) / √(x^3 + 1)
      = √(x^2 / (x^3 + 1) )
      = √( x^2/x^2 / (x^3/x^2 + 1/x^2) )
      = √( 1 / (x + 1/x^2) )
      As x goes to infinity, the denominator goes to infinity, so the whole fraction goes to zero and the square root of zero is zero, so f(x) goes to zero.

      f(x) = x / √(1 - x^3)
      = √( x^2 / (1 - x^3) )
      = √( 1 / (1/x^2 - x) )

      As x goes to infinity, the denominator goes to negative infinity, but we can't take the square root of a negative number, so f(x) doesn't have a limit as x goes to infinity.
      (5 votes)

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.