Using limθ→0 sin θ / θ = 1

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Question

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limx→0 (cos²x − cos x) / x²

Accepted Answer

Welcome back, everyone. Find the limit. Limit as x approaches 0 of c 2X minus s x divided by X2. We're going to begin solving this problem by assuming that our function is continuous at X equals 0. So let's substitute X equals 0, which gives us c 20 minus s0 divided by 02. Let's valuate the numerator. It gives us 1 minus 1, which is 0 in the numerator, divided by 0 in the denominator, and we can notice that this is an indeterminate form, so our function is not continuous at x equals 0. What we're going to do is basically rewrite our limit. In terms of cosine, let's recall that sequent of X is basically the reciprocal of cosine. So we have one divided by cosine of X. Our limit becomes limit of 1. Divided by cosine of X2 minus 1 divided by cosine of X divided by X2. What we also have to notice is that X approaches 0, meaning X is small, right? If x is small, we can use an approximation such that cosine of X is approximately equal to 1 minus X2 divided by 2. We're going to use this approximation to solve for the limit, and we're going to replace every cosine with 1 minus X2 divided by 2. So now, let's not that we want to valid one divided by cosine of X. This means that we are taking one. Divided by 1 minus X2 divided by 2, and we're going to divide, right? So we have 1 in the numerator, for the denominator, we're going to find the common denominator, which is 2, and this gives us 2 divided by 2. Minus X2. And now if we want to take the reciprocal we're basically flipping the bottom fraction, which gives us 2 divided by 2 minus X2. So now considering the limit, we end up with the limit as X approaches 0 of. 2 Divided by 2 minus X squared. Squared Minus 2 Divided by, let's add some space. And now we have 2 divided by 2 minus X squad, all of that. Divided by x 2. What we can do is perform factorization. We get limit as sex approaches 0, and basically we can factor out 2 divided by. 2 minus X2, which leaves us with 2 divided by 2 minus X2 minus 1, right? Notice that 2 divided by 2 minus X2 is the common factor. And all, all of that is divided by X2. Now, let's consider the part in C, we're going to get the limit, as X approaches 0 of 2 divided by 2 minus X2 multiplied by. 2 Divided by 2 minus X2, and now if we identify the common denominator, which is 2 minus X2, this means that we're subtracting 2 minus X2, which means that we're subtracting 2 and adding X2. All of that is divided by X2. Now 2 minus 2 gives us 0, and we can clearly see that we can now multiply, right, which is going to give us X squared. Next to the coefficient of 2, and then 2 minus X2 is going to become squared, right? Because we're multiplying it by itself. So we're going to add a square. And now we can just get rid of that factor because we have already multiplied. Now from here, let's understand that we are dividing that fraction by X2, so we can essentially cancel out X2. And we can rewrite the fraction as follows we have limit. As x approaches 0 of 2 divided by. 2 minus X2 squad. And now let's perform direct substitution once again. We get 2 divided by 2 minus 0 22, which is 22, and 2 divided by 2 squad is simply equal to 2 divided by 4, which is 1/2. And now this is our final answer. The limit to this problem is equal to 1/2. Thank you for watching.

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limx→0 (cos²x − cos x) / x²

Key Concepts

Limit of a Function

L'Hôpital's Rule

Trigonometric Limits

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