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

Limit properties

AP.CALC:
LIM‑1 (EU)
,
LIM‑1.D (LO)
,
LIM‑1.D.1 (EK)
,
LIM‑1.D.2 (EK)
What is the limit of the sum of two functions?  What about the product? Created by Sal Khan.

Want to join the conversation?

  • leaf green style avatar for user Themis
    I seen in my book the notation like lim f(x+y). Is this the same as f(x)+f(y) ?
    Also my teacher said that ln(x+y) is not lnx + lny but ln(x,y) = ln(x)+ln(y). Does ln represent a function?
    (2 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user Bryan Glesmann
    What I do not understand about limits is why you would want to do something like multiply or add them? How can limits be related to each other, let alone multiplied? Aren't limits simply in respect to themselves as a limit, and not in respect to other limits?
    (53 votes)
    Default Khan Academy avatar avatar for user
    • leafers ultimate style avatar for user KrisSKing
      These properties of limits give you tools for finding the limit of a more complicated function. You can break down a function into different parts (as defined by the limit properties) and find the limits of the different parts, putting them back together in the appropriate manner to find the limit of the overall function. For example the limit of:

      5x^3 + 4x^2 + 3x - 6

      can be found by breaking it up into 5 times (limit of x)^3 plus 4 times (limit of x)^2 plus 3 times (limit of x) - 6. So you only have to find the limit of one thing, x, and then apply the appropriate limit properties. This will become even more powerful when Sal introduces how to find the limit of a composite function.
      (2 votes)
  • starky ultimate style avatar for user mr.swedishfish
    Where can I find the "rigorous proof" of these properties?
    (10 votes)
    Default Khan Academy avatar avatar for user
    • mr pants teal style avatar for user Moon Bears
      A rigorous proof can usually be found in any old calculus text, in the section on limits. A fun exercise might be to write down the epsilon-delta definition of limits then try to figure out exactly how one would prove these statements!
      (7 votes)
  • starky seedling style avatar for user Cooper Maruyama
    At you wrote:
    lim f(x)g(x) = lim f(x) * g(x)
    This leads me to understand that lim ab = lim a * lim b

    However, at , you wrote:
    lim kf(x) = k * lim f(x)

    If lim ab = lim a * lim b, shouldn't lim kf(x) = lim k * lim f(x)?
    (4 votes)
    Default Khan Academy avatar avatar for user
  • piceratops ultimate style avatar for user grantgilmore6
    At , note that the lim of g(x) as it approaches c can not be 0. If it is, then the entire limit would not exist because a number divided by 0 is undefined.
    (10 votes)
    Default Khan Academy avatar avatar for user
    • piceratops ultimate style avatar for user jenvans6
      My real analysis textbook says that (and I will paraphrase) if the limits of f(x) and g(x) exist as x approaches c and the limit of g(x) as x approaches c is not equal to 0, THEN the limit of (f/g)(x)=(lim f(x))/(lim g(x)) as each of the limits approaches 0.

      I think this is what grant was getting at, but the video didn't go into detail about this.

      Also, Just Keith, your example with the limit is one where x approaches 0 and the answer is infinity, but if that was our g(x), it would not be the case that the limit of g(x) as x approaches c is 0. That said, it really does not apply to this situation. I see what your point is, but I think some of your response was fairly unrelated to the original comment.
      (2 votes)
  • leaf green style avatar for user popcorn.tomato.dude
    for the exponent property, why assume that the power is a rational number? would this still work if the power was sqrt(3) or something like that?
    (6 votes)
    Default Khan Academy avatar avatar for user
  • aqualine ultimate style avatar for user Bianca Otake
    Concerning , If you get the question "what is the product of the limit g(x) when x approaches c and the limit f(x) when x approaches c" given that f(x) is discontinuous at c. Would the answer be zero or just that the limit does not exist?
    (4 votes)
    Default Khan Academy avatar avatar for user
  • hopper cool style avatar for user anupamakundur23
    Is there any logarithm property for limits
    (4 votes)
    Default Khan Academy avatar avatar for user
    • aqualine ultimate style avatar for user Carleigh
      The closest thing to a 'logarithm property' is the rule regarding continuous functions. The limit of f(g(x)) is equal to f(the limit of g(x)), provided f is continuous at that limit. Logarithms are continuous on their domain, so we can apply that to say lim (ln(f(x))) = ln (lim f(x)) for a positive inner limit. We can also say ln(lim f(x)) = lim ln(f(x)), which is occasionally useful.
      (3 votes)
  • blobby green style avatar for user rosenthal
    I was given this problem in one of my textbooks about limits but do not understand the answer given. Find lim as h--->0 of (e^2(3+h)-e^2(3))/h by recognizing the limit as the definition of f (a′) for some function f and some value a.

    the answer given is f'(3) where f(x)=e^2x. f'(3)=2e^6 would appreciate any input on how they came to this answer

    thanks
    (2 votes)
    Default Khan Academy avatar avatar for user
    • piceratops ultimate style avatar for user Just Keith
      You don't plug in the value of x until after you have done the limit. So, you should do the limit calculation with the x instead of the 3. Once you have the limit in terms of x, then you plug in the x=3.

      Here is how to do the limit, if I understand the example correctly.
      f(x) = e^(2x)

      f'(x)= lim h→0 {e^[2(x+h)] - e^(2x) } / h
      f'(x) = lim h→0 {e^[2x+2h] - e^(2x) } / h
      f'(x) = lim h→0 {e^(2x)∙e^(2h)] - e^(2x) } / h
      Factor out e^(2x). Since it is not a function of h, we can factor it in front of the limit
      f'(x) = lim h→0 e^(2x) {e^(2h)] - 1 } / h
      f'(x) = e^(2x) lim h→0 {e^(2h) - 1 } / h
      -----
      Side calculation:
      Definition of e is lim h→0 (1+h)^(1/h)
      Thus, e^(2h) = lim h→0 [ (1+h)^(1/h)]^(2h)
      e^(2h)= lim h→0 (1+h)^(2h/h)
      e^(2h)= lim h→0 (1+h)^(2)
      e^(2h)= lim h→0 (1 + 2h + h²)
      Since we both limits have the limit variable approaching 0, we can substitute the e^(2h) in main calculation with (1 + 2h + h²)
      ------
      Back to main calculation:
      f'(x) = e^(2x) lim h→0 {e^(2h) - 1 } / h
      f'(x) = e^(2x) lim h→0 { 1 + 2h + h² - 1 } / h
      f'(x) = e^(2x) lim h→0 {2h + h² } / h
      f'(x) = e^(2x) lim h→0 h(2 + h) / h
      cancel out the h
      f'(x) = e^(2x) lim h→0 (2 + h)
      Apply the limit:
      f'(x) = e^(2x) (2 + 0)
      f'(x) = e^(2x) (2)
      f'(x) = 2e^(2x)
      Now, apply x=3
      f'(3) = 2e^(2∙3) = 2e^6
      (7 votes)
  • blobby green style avatar for user Montana Burr
    What is the limit of f(x) raised to the g(x) power?
    (4 votes)
    Default Khan Academy avatar avatar for user

Video transcript

What I want to do in this video is give you a bunch of properties of limits. And we're not going to prove it rigorously here. In order to have the rigorous proof of these properties, we need a rigorous definition of what a limit is. And we're not doing that in this tutorial, we'll do that in the tutorial on the epsilon delta definition of limits. But most of these should be fairly intuitive. And they are very helpful for simplifying limit problems in the future. So let's say we know that the limit of some function f of x, as x approaches c, is equal to capital L. And let's say that we also know that the limit of some other function, let's say g of x, as x approaches c, is equal to capital M. Now given that, what would be the limit of f of x plus g of x as x approaches c? Well-- and you could look at this visually, if you look at the graphs of two arbitrary functions, you would essentially just add those two functions-- it'll be pretty clear that this is going to be equal to-- and once again, I'm not doing a rigorous proof, I'm just really giving you the properties here-- this is going to be the limit of f of x as x approaches c, plus the limit of g of x as x approaches c. Which is equal to, well this right over here is-- let me do that in that same color-- this right here is just equal to L. It's going to be equal to L plus M. This right over here is equal to M. Not too difficult. This is often called the sum rule, or the sum property, of limits. And we could come up with a very similar one with differences. The limit as x approaches c of f of x minus g of x, is just going to be L minus M. It's just the limit of f of x as x approaches c, minus the limit of g of x as x approaches c. So it's just going to be L minus M. And we also often call it the difference rule, or the difference property, of limits. And these once again, are very, very, hopefully, reasonably intuitive. Now what happens if you take the product of the functions? The limit of f of x times g of x as x approaches c. Well lucky for us, this is going to be equal to the limit of f of x as x approaches c, times the limit of g of x, as x approaches c. Lucky for us, this is kind of a fairly intuitive property of limits. So in this case, this is just going to be equal to, this is L times M. This is just going to be L times M. Same thing, if instead of having a function here, we had a constant. So if we just had the limit-- let me do it in that same color-- the limit of k times f of x, as x approaches c, where k is just some constant. This is going to be the same thing as k times the limit of f of x as x approaches c. And that is just equal to L. So this whole thing simplifies to k times L. And we can do the same thing with difference. This is often called the constant multiple property. We can do the same thing with differences. So if we have the limit as x approaches c of f of x divided by g of x. This is the exact same thing as the limit of f of x as x approaches c, divided by the limit of g of x as x approaches c. Which is going to be equal to-- I think you get it now-- this is going to be equal to L over M. And finally-- this is sometimes called the quotient property-- finally we'll look at the exponent property. So if I have the limit of-- let me write it this way-- of f of x to some power. And actually, let me even write it as a fractional power, to the r over s power, where both r and s are integers, then the limit of f of x to the r over s power as x approaches c, is going to be the exact same thing as the limit of f of x as x approaches c raised to the r over s power. Once again, when r and s are both integers, and s is not equal to 0. Otherwise this exponent would not make much sense. And this is the same thing as L to the r over s power. So this is equal to L to the r over s power. So using these, we can actually find the limit of many, many, many things. And what's neat about it is the property of limits kind of are the things that you would naturally want to do. And if you graph some of these functions, it actually turns out to be quite intuitive.