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

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  • 0:00 Steps to Solve Limit…
  • 4:09 Solution to Finding the Limit
  • 4:50 L'Hopital's Rule,…
  • 7:11 Lesson Summary
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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.



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?


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.


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))


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.


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.

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