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.

llimx→0 (x −x cos x) / sin² 3x

Accepted Answer

Welcome back, everyone. Determine the following limit. Limit SX approaches 0 of 1 minus cosine of 3 x divided by sin squared of 2 X. Let's consider this limit. If we simply perform a direct substitution of X equals 0, our numerator is going to become 1 minus cosine of 0, which is 1 minus 1. And that's 0 and our denominator is going to be sin squared of 0, which is 0 as well. So we have an indeterminate form of 0 divided by 0, right, and this means that we need to evaluate this limit using a different method than just substitution. What we're going to do in this problem is simply refo that cosine of X can be approximated as 1 minus X2 divided by 2 if X is small. And we know that x approaches 0 looking at the limit, so we can use this approximation. And also we can say that sin of x is approximately equal to X with the same condition if X approaches 0. Therefore, because we have cosine of 3 X, we can approximate it as 1 minus 3 x 2 divided by 2. And this gives us 1 minus 9 X 2 divided by 2, right? And now, considering our denominator, we can sell that sign of 2 X is approximately equal to 2 X. And now let's use those values in our limit. We get limit as X approaches 0 of 1 minus 1 minus 9 X squad. Divided by 2, and for our Denominator. We're going to use sine squared of 2 x approximation. So, sign of 2 X is approximately equal to 2 X, we basically have to square 2 X. Let's continue. We get limit as X approaches 0 of 1 minus 1, that's 0, negative of 9 X 2 divided by 2 makes it positive. And then we are dividing by 4 X2. And now we essentially have a double division, right? 9 X 2 divided by 2 divided by or X2, gives us 8 X squad in the denominator. We can combine our denominators. And we can see from here that X2d can be canceled out, right? And then we are evaluating the limit of a constant which is equal to that constant itself because it's independent of X. So the answer is 9 divided by 8. That's our final answer, and thank you for watching.

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

llimx→0 (x −x cos x) / sin² 3x

Key Concepts

Limit of a Function

Trigonometric Limits

Indeterminate Forms

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