If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

Analyzing unbounded limits: rational function

AP.CALC:
LIM‑2 (EU)
,
LIM‑2.D (LO)
,
LIM‑2.D.1 (EK)
,
LIM‑2.D.2 (EK)
Sal analyzes the behavior of f(x)=-1/(x-1)² around its asymptote at x=1.

Want to join the conversation?

  • spunky sam green style avatar for user Taha Anouar
    can we say in that case the limit as x approach 1 will be negative infinity.. because the left and the right limit are the same ??
    (7 votes)
    Default Khan Academy avatar avatar for user
    • female robot grace style avatar for user tyersome
      We can (and often do) say that, but to be completely correct we should say something like "the limit as x approaches 1 diverges towards negative infinity".

      The reason we aren't supposed to say a limit will be (negative) infinity, is that infinity is not a real number – it might be better thought of as a direction ...

      If you are interested, there is a whole area of mathematics devoted to infinities!
      (19 votes)
  • aqualine ultimate style avatar for user George Evans
    Throughout the video, is it incorrect for it to be saying that the limit of the function = positive or negative infinity? I ask because thinking the way it talks in this video made me think that both limits talked about in this video exist, which Sal says is wrong:
    https://www.khanacademy.org/math/differential-calculus/dc-limits/dc-point-continuity/v/continuity-at-a-point-graphically
    (2 votes)
    Default Khan Academy avatar avatar for user
    • mr pants teal style avatar for user Moon Bears
      Unbounded limits don't exist; however, they are different from limits such as a_n = (-1)^n ; this sequence doesn't have a limit merely because it is alternating between 1 & -1, though its absolute value stays at 1. Unbounded limits aren't oscillating - they keep getting bigger or smaller. So we define infinity & - infinity to represent that. Technically they aren't "real numbers" but they are apart of the extended real number system. A reason as to why the limits can't exist is because consider 1 = x*1/x (x > 0) as x approaches 0 from the right. If the limit existed we could write lim x * 1/x = lim x * lim 1/x = 0 * (infinity) = 0. But the limit is clearly 1. So saying the limit doesn't exist is just a reminder we can't use limit properties to pull apart operations.
      (1 vote)
  • leaf green style avatar for user Moly
    how can i know the limit as x approaches 1 in the above function ?
    (1 vote)
    Default Khan Academy avatar avatar for user
  • aqualine seed style avatar for user Mariem Bakr
    is there any other way to do it except the estimating tabls way
    (0 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user Mais Banna
    Can I put a function and then you give me the answers of its limits step by step?
    (0 votes)
    Default Khan Academy avatar avatar for user
  • winston baby style avatar for user joshualewin
    How can you put a sign on infinity? I can prove you can't given 1/∞=0.
    x=x
    x=x*x/x
    x=(x^2)(1/x)
    |x|=|x^2||1/x|
    |x|=(x^2)|1/x|
    |x|/x=|1/x|(x^2)/x
    |x|/x=|1/x|x
    |x|/x=|1/x|/(1/x)
    Now substitute +-∞. You get:
    ∞/(+-∞)=|1/∞|/(1/+-∞)
    We know that 1/∞=0, so 1/0=∞, so:
    ∞/(+-∞)=0/+-0
    ∞/(+-∞)=0/0
    ∞/(+-∞)=(1/∞)/(1/∞)
    ∞/(+-∞)=∞/∞
    (+-∞)/∞=∞/∞
    +-∞=∞
    Positive and negative infinity are both actually just plain old infinity. Right?
    (0 votes)
    Default Khan Academy avatar avatar for user

Video transcript

- [Voiceover] Let f of x be equal to negative one over x minus one squared. Select the correct description of the one-sided limits of f at x equals one. And so we can see, we have a bunch of choices where we're approaching x from the right-hand side and we're approaching x from the left-hand side. And we're trying to figure out do we get unbounded on either of those, in the positive, towards positive infinity or negative infinity. And there's a couple of ways to tackle it. The most straightforward, well, let's just consider each of these separately. So we can think about the limit of f of x as x approaches one from the positive direction and limit of f of x as x approaches one, as x approaches one from the left-hand side. This is from the right-hand side. This is from the left-hand side. So I'm just gonna make a table and try out some values as we approach, as we approach one from the different sides, x, f of x, and I'll do the same thing over here. So, we are going to have our x and have our f of x and if we approach one from the right-hand side here, that would be approaching one from above, so we could try 1.1, we could try 1.01. Now f of 1.1 is negative one over 1.1 minus one squared. So see this denominator here is going to be .1 squared. So this is going to be, this is going to be 0.01, and so this is going to be negative 100. So let me just write that down. That's going to be negative 100. So if x is 1.01, well, this is going to be negative one over 1.01 minus one squared. Well, then this denominator this is going to be, this is the same thing as 0.01 squared, which is the same thing as 0.0001, 1/10000. And so the negative one 1/10000 is going to be negative 10,000. So, let's just write that down, negative 10,000. And so this looks like, as we get closer, 'cause notice, as I'm going here I am approaching one from the positive direction, I'm getting closer and closer to one from above and I'm going unbounded towards negative infinity. So this looks like it is negative infinity. Now we can do the same thing from the left-hand side. I could do 0.9, I could do 0.99. Now 0.9 is actually also going to get me negative 100 'cause 0.9 minus one is going to be negative .1 but then when you square it the negative goes away so you get a .01 and then one divided by that is 100 but you have the negative, so this is also negative 100. And if you don't follow those calculations, I'll do it, let me do it one more time just so you see it clearly. This is going to be negative one over, so now I'm doing x is equal to 0.99, so I'm getting even closer to one, but I'm approaching from below from the left-hand side. So this is going to be 0.99 minus one squared. Well, 0.99 minus one is, is going to be negative 1/100, so this is going to be negative 0.01 squared. When you square it the negative goes away and you're left with 1/10000. So this is going to be 0.0001 and so when you evaluate this you get 10,000. So that, or sorry, you get negative 10,000. So in either case, regardless of which direction we approach from, we are approaching negative infinity. So that is this choice right over here. Now there's other ways you could have tackled this if you just look at, kind of, the structure of this expression here, the numerator is a constant, so that's clearly always going to be positive. Let's ignore this negative for the time being. That negative's out front. This numerator, this one is always going to be positive. Down here, we're taking at x equals one, while this becomes zero and the whole expression becomes undefined, but as we approach one, x minus one could be positive or negative as we see over here, but then when we square it, this is going to become positive as well. So the denominator is going to be positive for any x other than one. So positive divided by positive is gonna be positive but then we have a negative out front. So this thing is going to be negative for any x other than one, and it's actually not defined at x equals one. And so you could, from that, you could deduce, well, okay then, we can only go to negative infinity there's actually no way to get positive values for this function.