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

In this video, we explore the limit of (1-cos(x))/x as x approaches 0 and show that it equals 0. We use the Pythagorean trigonometric identity, algebraic manipulation, and the known limit of sin(x)/x as x approaches 0 to prove this result. This concept is helpful for understanding the derivative of sin(x).

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• 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.
• I thought about it too, but then we should realize that what we did was trial and error method of figuring out what it could be, but the above method was something more formal and acceptable.
• i want to practice this one. Where to find good exercise?
• 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.
• 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.
• 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.
• what if the limit is approaching infinity
• What do you think will happen?

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→∞?

• 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.
• Precalculus unit 2 is good for trig. It talks about all the trig identities.

You can also use the whole Trigonometry Course.

Another option is to go to SAT section and practice some trig there. They give real examples and can give you a firm understanding.
(1 vote)
• 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...
• i suppose we need that the other limit is at least defined, but that's given.
• How to graph it?
• 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