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Limits at infinity of quotients (Part 2)

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 analyzes the limits at infinity of three different rational functions. He finds there are three general cases of how the limits behave. Created by Sal Khan.

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  • aqualine ultimate style avatar for user Nikita
    I understand how he gets to the limits and all, but can someone give me a 100% accurate definition of what exactly a "asymptote" is? what is an asymptote? Is it just the graph of the limit? please help me !
    (41 votes)
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    • leaf green style avatar for user patticus
      This is actually not quite accurate.

      Technically an asymptote to a curve is a line such that the distance between the curve and the line approaches zero as they both tend toward infinity.

      A function can never cross a vertical asymptote, but it may cross a horizontal asymptote an infinite number of times (thus reaching it an infinite number of times). For example observe the limit of sin(x)/x as x approaches infinity.
      (21 votes)
  • blobby green style avatar for user luckysidgem720
    Are + and - infinity like approaching from the positive and negative sides? So limits approaching infinity dont exist, because + and - infinity yield different values?
    (8 votes)
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    • male robot donald style avatar for user Jeremy
      Not exactly,

      positive and negative infinity represent the opposite "ends" of the number line. And here, "ends" is in quotation marks because the number line NEVER actually ends, it goes on forever in both directions. Basically positive infinity means to keep going towards bigger and bigger positive numbers. Think of the biggest positive number you can think of, and then go even bigger than that… and keep doing that… FOREVER! That's positive infinity.

      For negative infinity, think of the most negative number you can think of, and then think of an even more negative number, and keep doing that, FOREVER.

      So you see, if a limit approaches positive infinity from one side, and negative infinity from the other side… it doesn't approach the same thing from both sides. THIS is why the limit doesn't exist. It would be the same as saying that a limit that approaches 3 from the positive side and 2 from the negative side also doesn't exist. In order for a limit to exist it must approach the same thing from both sides.
      (27 votes)
  • old spice man green style avatar for user Petrie (Peter S. Asiain III)
    Where exactly can we use the concept of limits in our life?
    In what kind of situations in our life do we need to find limits?
    Please give me a real life example where in we have to find the limit.
    Thanks in advance!
    (7 votes)
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    • leafers ultimate style avatar for user Muhannad AlAyoubi
      Limits are not that helpful in everyday life. However, as you study Calculus, you'll see that every single concept in Calculus in based upon limits. Derivatives are defined using limits, we can find the area under a curve, volume, surface area, arc length, radius of curvature using integrals that are all based upon limits. Without limits, there is no Calculus.
      (5 votes)
  • spunky sam blue style avatar for user Cam
    So then would it be safe to assume that any time the numerator is growing faster than the denominator it will equal infinity, and any time the denominator is growing faster than the numerator it will equal 0? Or is there an exception?
    (11 votes)
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  • male robot hal style avatar for user Logan LaFollette
    At how can he simplify that to 1/2x when there are two different powers?
    (6 votes)
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  • piceratops ultimate style avatar for user mark.ninosque75
    at , is the simplification of (4/250)x to (2/125)x valid ?
    (3 votes)
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    • piceratops ultimate style avatar for user Just Keith
      Yes, but it is unnecessary. With limits, since you often have them diverge toward +∞ or −∞ or else tend toward 0, you can save yourself unnecessary work by not simplifying any constants until you know you don't have an infinity or zero situation. When tending toward 0, your constant is irrelevant and there is no need to simplify. When tending toward ∞, you need only determine the sign of your constant, to determine whether you're tending toward + or − infinity.

      It is only when you're tending toward a non-zero but finite number that you need to simplify your constant.

      Note: The above applies to real numbers. If imaginary numbers get involved, the considerations can be more complicated.
      (7 votes)
  • blobby green style avatar for user Asad Rehman
    Lim(x--->infinity) (1+1/n)^n =e
    Lim(x--->infinity) (1+1/x)^1/x=e
    How to do solve these very strange infinity limits or either to prove them they equal to e?
    (3 votes)
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    • purple pi purple style avatar for user redthumb.liberty
      x		(1 + 1/x)^x	(1 + 1/x)^(1/x)
      1 2.000000000 2.00000000000000
      10 2.593742460 1.00957658300000
      100 2.704813829 1.00009950825915
      1000 2.716923932 1.00000099950083
      10000 2.718145927 1.00000000999950
      100000 2.718268237 1.00000000010000
      1000000 2.718280469 1.00000000000100
      10000000 2.718281694 1.00000000000001
      (6 votes)
  • leaf yellow style avatar for user Narek Kazarian
    In the solutions manual of my Calculus textbook, it gets the answer using a slightly different method. It divides like every term in the numerator and the denominator by the highest degree i guess and does all these weird calculations and then gets the answer. I understand this method much easier though (getting the highest power of both the numerator and denominator and then applying the x-> component) I guess my question is if I do this method that you are teaching on a test, is it still valid and legitimate? Of will i have to provide more work to justify my answer? Sorry for the incredibly long comment. Thanks.
    (2 votes)
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    • spunky sam blue style avatar for user Ethan Dlugie
      Sal's method is more of a "rationalize it" (no pun intended) approach. What your textbook says is the method I prefer, mainly because it gives you a clear, algebraic way to evaluate the limit.
      As for which method to use on a test, that is something you should ask your teacher.
      (8 votes)
  • male robot hal style avatar for user Enn
    Will a similar method be used for an expression in which there is no denominator like for limit x tending to infinity √(n-1) - √n
    Here will it be correct to say that this is the same as √n - √n (As the 1 will be negligible when n tends to infinity) and thus the limit equals 0 ?
    (3 votes)
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    • leafers seed style avatar for user Travis Bartholome
      Yep. That general pattern of thinking is correct; actually, though, the example you've mentioned is a bit interesting in the sense that we can't always define a limit that takes the form ∞ - ∞. In this case, because the two terms are of the same degree, the limit is equal to 0 (and a quick glance at the graph of y = sqrt(x-1) - sqrt(x) confirms that as x approaches infinity, y approaches 0). As you said, it resembles y = sqrt(x) - sqrt(x) = 0 in the limit.

      Other limits of a similar nature may not always behave the same way. Take the limit of x^3 - x^2 as x approaches infinity, and we get infinity rather than 0 because the terms are of a different degree (which seems fairly clear just by looking at the function). Sometimes the examples are less clear-cut, so it's worth exercising some caution with limits of the form ∞ - ∞.

      I hope you find that helpful; you're definitely on the right track.
      (5 votes)
  • blobby green style avatar for user Ashley Smith St. Clair
    When dealing with these questions where x approaches infinity or negative infinity, is it possible to get an actual number, or will the answer always be infinity, negative infinity or zero?
    (1 vote)
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Video transcript

Let's do a few more examples of finding the limit of functions as x approaches infinity or negative infinity. So here I have this crazy function. 9x to the seventh minus 17x to the sixth, plus 15 square roots of x. All of that over 3x to the seventh plus 1,000x to the fifth, minus log base 2 of x. So what's going to happen as x approaches infinity? And the key here, like we've seen in other examples, is just to realize which terms will dominate. So for example, in the numerator, out of these three terms, the 9x to the seventh is going to grow much faster than any of these other terms. So this is the dominating term in the numerator. And in the denominator, 3x to the seventh is going to grow much faster than an x to the fifth term, and definitely much faster than a log base 2 term. So at infinity, as we get closer and closer to infinity, this function is going to be roughly equal to 9x to the seventh over 3x to the seventh. And so we can say, especially since, as we get larger and larger as we get closer and closer to infinity, these two things are going to get closer and closer each other. We could say this limit is going to be the same thing as this limit. Which is going to be equal to the limit as x approaches infinity. Well, we can just cancel out the x to the seventh. So it's going to be 9/3, or just 3. Which is just going to be 3. So that is our limit, as x approaches infinity, in all of this craziness. Now let's do the same with this function over here. Once again, crazy function. We're going to negative infinity. But the same principles apply. Which terms dominate as the absolute value of x get larger and larger and larger? As x gets larger in magnitude. Well, in the numerator, it's the 3x to the third term. In the denominator it's the 6x to the fourth term. So this is going to be the same thing as the limit of 3x to the third over 6x to the fourth, as x approaches negative infinity. And if we simplified this, this is going to be equal to the limit as x approaches negative infinity of 1 over 2x. And what's this going to be? Well, if the denominator, even though it's becoming a larger and larger and larger negative number, it becomes 1 over a very, very large negative number. Which is going to get us pretty darn close to 0. Just as 1 over x, as x approaches negative infinity, gets us close to 0. So this right over here, the horizontal asymptote in this case, is y is equal to 0. And I encourage you to graph it, or try it out with numbers to verify that for yourself. The key realization here is to simplify the problem by just thinking about which terms are going to dominate the rest. Now let's think about this one. What is the limit of this crazy function as x approaches infinity? Well, once again, what are the dominating terms? In the numerator, it's 4x to the fourth, and in the denominator it's 250x to the third. These are the highest degree terms. So this is going to be the same thing as the limit, as x approaches infinity, of 4x to the fourth over 250x to the third. Which is going to be the same thing as the limit of-- let's see, 4, well I could just-- this is going to be the same thing as-- well we could divide two hundred and, well, I'll just leave it like this. It's going to be the limit of 4 over 250. x to the fourth divided by x to the third is just x. Times x, as x approaches infinity. Or we could even say this is going to be 4/250 times the limit, as x approaches infinity of x. Now what's this? What's the limit of x as x approaches infinity? Well, it's just going to keep growing forever. So this is just going to be, this right over here is just going to be infinity. Infinity times some number right over here is going to be infinity. So the limit as x approaches infinity of all of this, it's actually unbounded. It's infinity. And a kind of obvious way of seeing that, right, from the get go, is to realize that the numerator has a fourth degree term. While the highest degree term in the denominator is only a third degree term. So the numerator is going to grow far faster than the denominator. So if the numerator is growing far faster than the denominator, you're going to approach infinity in this case. If the numerator is growing slower than the denominator, if the denominator is growing far faster than the numerator, like this case, you are then approaching 0. So hopefully you find that a little bit useful.