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.

limθ→0 (1 − cos θ) / sin 2θ

Accepted Answer

Welcome back, everyone. Determine the limit. Limit as theta approaches 0 of 1 minus cosine of 4 theta divided by theta multiplied by sin of 6 theta. For this problem, if we perform a direct substitution of theta equals 0, we're going to get 1 minus cosine of 0, which is 1 minus 1 and that's 0 in the numerator, and for the denominator, we get 0 as well, right? So 0 divided by 0 is an indeterminate form. So we cannot really evaluate that limit by direct substitution, and what we're going to do is simply notice that theta approaches 0, right? This means that theta is small. So we're going to use approximations for this problem. First of all, let's recall that cosine of X can be approximated as 1 minus X2 divided by 24 small values of x. And now sine of x. Can be approximated as X if X is small, right? So in this context, we can say that cosine. Of 4 theta is approximately equal to 1 minus 4 quad divided by 2, right, using our formula, and we can now simplify, and this is equal to 1 minus 16 the quad divided by 2, which gives us 1 minus 8. Theta squared. And now let's consider our denominator, so we have Sign. Let's begin with our sign function. Of 6 theta. Which is approximately equal to the angle 6 theta, right? Now what we're going to do is simply use our limit. We have limit as theta approaches 0, and we're going to replace cosine and sign with the approximations. This gives us 1 minus 1 minus 8 theta squared in the numerator, and now for the denominator, we end up with theta. Multiplied by 6 theta. And now let's simplify. We're going to get the limit as theta approaches 0 of 1 minus 1 minus. 8 theta squared, right? So if we distribute the terms, we're going to get negative of -8 theta squad, which is 8 the squad. And this is divided by 6 theta squared. So we can now clearly see that data squared can be canceled out and we get the limit of a constant, which is 8 divided by 6, right? Because it's independent of data, we can just write our limit as the constant itself, and then we can simply use the simplification method in which we divide both sides of the fraction by the greatest common factor. In this case, we're dividing both sides by 2, which gives us 4/3 as the final answer to this limit. Thank you for watching.

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limθ→0 (1 − cos θ) / sin 2θ

Key Concepts

Limit of a Function

Trigonometric Limits

L'Hôpital's Rule

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