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Limit of (1-cos(x))/x as x approaches 0

AP.CALC:
LIM‑1 (EU)
,
LIM‑1.E (LO)
,
LIM‑1.E.2 (EK)
Showing that the limit of (1-cos(x))/x as x approaches 0 is equal to 0. This will be useful for proving the derivative of sin(x).

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  • aqualine ultimate style avatar for user Jeffrey Hu
    Would the following proof also work?
    Proof:
    Note that 1-cos(x)>0 for all x such that x is not equal to 0.
    As x approaches 0 from the negative side, (1-cos(x))/x will always be negative. As x approaches 0 from the positive side, (1-cos(x))/x will always be positive. We know that the function has a limit as x approaches 0 because the function gives an indeterminate form when x=0 is plugged in. Therefore, because the limit from one side is positive and the limit from the other side is negative, the limit must be 0.
    (42 votes)
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  • leafers seedling style avatar for user Solongo Batbayar
    i want to practice this one. Where to find good exercise?
    (7 votes)
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  • blobby green style avatar for user Pratyush Jethwani
    At , the instructor says, " Well here, the limit of the product of these two expressions, is going to be the same thing as the product of the limits." I don't understand how that works. Can you do that with all limits.
    (5 votes)
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  • blobby green style avatar for user josefthink
    If you are saying at the end that cos(x)=1, why is the whole manipulation needed? You can simply say that 1-1=0? And the denominator is not important at all.
    (5 votes)
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    • female robot grace style avatar for user loumast17
      The denominator is important. Say instead of this you had something like (x-2)/x and want the limit as x goes to 0. Here the limit is not 0, or -2 if you simply plugged 0 in for x. It just happened that this one worked out like that. So the moral of the story is if you have a 0 in the denominator, you're going to want to manipulate your function so you don't.
      (5 votes)
  • blobby green style avatar for user Araceli Serrano
    what if the limit is approaching infinity
    (1 vote)
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    • female robot grace style avatar for user tyersome
      What do you think will happen?

      To answer this ask yourself:
      What are the maximum and minimum values that 1-cos(x) can take?

      What happens when I divide those values by increasingly large values as x→∞?

      Does this help you answer your question?
      (4 votes)
  • blobby green style avatar for user Ayshi
    How to graph it?
    (2 votes)
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    • leaf green style avatar for user cossine
      First sketch 1-cos x then x. Determine where functions 1-cos x and x are positive and negative to determine where (1-cos x)/x will be positive and negative. Find any asymptotes (x=0). To help sketch determin whether the function is odd and even.

      If required check for concavity using the second derivative as well as max and minimums
      (2 votes)
  • blobby green style avatar for user cscambler
    Why do we need that the limit as x goes to 0 of sin x/ x is 1 for this? isn't it enough that the limit as x goes to 0 of sin x / 1 + cos x = 0? That's enough to make the whole product 0 right...
    (2 votes)
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  • starky tree style avatar for user Liam
    When he multiplies (1-cosx)/x at the beginning by (1+cosx)/(1+cosx) (Timestamp ), the new expression is now undefined when cosx = -1. Whenever you multiply by anything that introduces undefined values that weren't in the original expression, should you be worried?
    (2 votes)
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  • male robot johnny style avatar for user Mohamed Ibrahim
    Can you graph this function without using a calculator ?
    (1 vote)
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    • blobby green style avatar for user Seth
      Yes, "curve sketching" is explored later in this course. This function in particular is going to be pretty tricky because it requires a good understanding of the sine and cosine functions, but you can get a pretty good estimate of what it looks like by finding its first and second derivatives and by using curve sketching techniques.
      (2 votes)
  • blobby green style avatar for user Tyler Christianson
    I'm struggling with any lesson that uses Trigonometric Identities because I haven't used them in so long. I'd prefer to not go back and do all of the Trigonometry course, but I was wondering what are the best lessons I could review to help me get caught up to speed with all the trigonometric identities manipulation they use in the video? Thank you.
    (1 vote)
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Video transcript

- [Instructor] What we wanna do in this video is figure out what the limit, as x approaches zero, of one minus cosine of x over x is equal to. And we're going to assume we know one thing ahead of time. We're going to assume we know that the limit, as x approaches zero, of sine of x over x, that this is equal to one. Now, I'm not gonna reprove this in this video, but we have a whole other video dedicated to proving this famous limit, and we do it using the squeeze, or the sandwich theorem. So, let's see if we can work this out. So the first thing we're going to do is algebraically manipulate this expression. What I'm going to do is I'm going to multiply both the numerator and the denominator by one plus cosine of x. So, times... The denominator I have to do the same thing, one plus cosine of x. I'm not changing the value of the expression, this is just multiplying it by one. What does that do for us? Well I can rewrite the whole thing as the limit, as x approaches zero, so one minus cosine of x times one plus cosine of x, well that is just going to be... Put this in another color. That is going to be one squared, which is just one, minus cosine squared of x. Cosine squared of x, difference of squares. And then in the denominator, I am going to have these, which is just x times 1 plus cosine of x. Now what is one minus cosine squared of x? Well, this comes straight out of the Pythagorean identity, trig identity. This is the same thing as the sine squared of x. So, sine squared of x. And so, I can rewrite all of this as being equal to the limit, as x approaches zero, and let me rewrite this as... Instead of sine squared of x, that's the same thing as sine of x times sine of x. Let me write it that way. Sine x times sine x. So, I'll take the first sine of x, so I'll take this one right over here, and put it over this x. So, sine of x over x times the second sine of x, this one, over one plus cosine of x times sine of x over one plus cosine of x. All I've done is I've leveraged a trigonometric identity, and I've done a little bit of algebraic manipulation. Well here, the limit of the product of these two expressions, is going to be the same thing as the product of the limits. So I can rewrite this as being equal to the limit, as x approaches zero, of sine of x over x times the limit, as x approaches zero, of sine of x over one plus cosine of x. Now, we said, going into this video, that we're going to assume that we know what this is. We've proven it in other videos. What is the limit, as x approaches zero, of sine of x over x? Well, that is equal to one. So, this whole limit is just going to be dependent on whatever this is equal to. Well, this is pretty straight forward, here. As x approaches zero, the numerator's approaching zero, sine of zero is zero. The denominator is approaching... Cosine of zero is one, so the denominator is approaching two. So this is approaching zero over two, or just zero. That's approaching zero. One times zero, well this is just going to be equal to zero. And we're done. Using that fact, and a little bit of trig identities, and a little bit of algebraic manipulation, we were able to show that our original limit, the limit, as x approaches zero, of one minus cosine of x over x is equal to zero. And I encourage you to graph it. You will see that that makes sense from a graphical point of view, as well.