Copyright

Finding the Limit of (1-cos(x))/x Video

An error occurred trying to load this video.

Try refreshing the page, or contact customer support.

Coming up next: Velocity and the Rate of Change

You're on a roll. Keep up the good work!

Take Quiz Watch Next Lesson
 Replay
Your next lesson will play in 10 seconds
  • 0:00 Steps to Solve Limit…
  • 4:09 Solution to Finding the Limit
  • 4:50 L'Hopital's Rule,…
  • 7:11 Lesson Summary
Save Save Save

Want to watch this again later?

Log in or sign up to add this lesson to a Custom Course.

Log in or Sign up

Timeline
Autoplay
Autoplay
Speed Speed

Recommended Lessons and Courses for You

Lesson Transcript
Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

We will see how to find the limit of (1 - cos(x)) / x, as x approaches a, for the case when a is 0 and when a is not equal to zero. We will see two different methods for when a is 0, so you can choose which method you like better.

Steps to Solve Limit of 1-Cos(x)/x

When it comes to finding the limit of a function, as x approaches some value a, there are many different methods that can be attempted. Depending on the function, some of these methods will work, and some won't. We are looking to find the limit of (1-cos(x)) / x, as xa.


lim1cosxx1


lim1cosxx2


To do this, we use two different methods depending on the value of a. One is for when a = 0, and the other is for when a ≠ 0. First, let's look at when a ≠ 0.

When a ≠ 0, finding the limit of (1 - cos(x)) / x is really quite easy. We use the plug-in method, which involves simply plugging a into (1 - cos(x)) / x for x.

We see that when a ≠ 0, we get that the limit of (1 - cos(x)) / x, as xa, is (1 - cos(a) / a. Pretty easy and straightforward, wouldn't you say?


lim1cosxx8


Now let's consider when a = 0. When this is the case, it's a little trickier to find this limit. The reason for this is because if we try to use the plug-in method, we end up with a zero in the denominator, and one of the number one rules in mathematics is that we cannot divide by zero.

Therefore, when this is the case, we use a well-known theorem that states that the limit of sin(x) / x, as x → 0, is 1.


lim1cosxx3


You may be wondering how this theorem has anything to do with our function. Well, as it turns out, we can manipulate the function (1 - cos(x)) / x by multiplying both the numerator and the denominator by 1 + cos(x) and using the trigonometric identity sin2 (x) = 1 - cos2 (x), we end up with a function involving sin(x) / x. In doing this, we turn the problem into taking the limit of the following:

(sin(x) / x)*(sin(x) / (1+ cos(x))


lim1cosxx5


Now we use the product rule for limits, which states that the limit of a product of functions is the product of the limits of the functions. Thus, we multiply the limit of sin(x) / x, as x → 0, which is 1, times the limit of sin(x) / (1 + cos(x)), as x → 0, which is 0.

Therefore, all together, we have that the limit of (1 - cos(x)) / x, as x → 0, is 0.

Solution to Finding the Limit

When a ≠ 0, we get that the limit of (1 - cos(x)) / x, as xa, is (1 - cos(a)) / a, and when a = 0, we get that the limit of (1 - cos(x)) / x, as x → 0, is 0.


lim1cosxx6


L'Hopital's Rule, Alternate Method

We've seen how to find this limit for the different possible values of a. It's fairly obvious that the process is much more involved when a = 0. Well, I have good news! If you didn't like the process we just went through, there is another method that we can use to find the limit when a = 0, and it's called L'Hopital's Rule.

L'Hopital's Rule, named after 17th century French mathematician, Guillaume de l'Hopital, can be used to find limits when the plug-in method results in indeterminate forms 0/0 or ∞ / ∞. We saw that if we try to use the plug-in method on the limit of (1 - cos(x)) / x, as x → 0, we get the indeterminate form 0/0. Therefore, we can use this rule for this limit, so let's figure out how.

To unlock this lesson you must be a Study.com Member.
Create your account

Register to view this lesson

Are you a student or a teacher?

Unlock Your Education

See for yourself why 30 million people use Study.com

Become a Study.com member and start learning now.
Become a Member  Back
What teachers are saying about Study.com
Try it risk-free for 30 days

Earning College Credit

Did you know… We have over 200 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Transferring credit to the school of your choice

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

Create an account to start this course today
Try it risk-free for 30 days!
Create an account
Support