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.

limh→0− h / sin 3h

Accepted Answer

Welcome back, everyone. Find the limit of the function F of T equals -2 T divided by sin of 4 T S T approaches 0. We're given 4 answer choices A says -1/4, B 1/4, C -12, and D 12. For this problem, let's simply recall that the limit as X approaches 0 of X divided by sin of X is equal to 1 by definition. And we're going to use this definition to manipulate the limit that we want to evaluate, right? So we have limit as T approaches 0 of -2 T divided by sin of 4 T. Now let's focus on the sign. Sine has an angle of 4 T, right, but our numerator has TT. So what we can do is simply rewrite the limit as limit as T approaches 0 of -4T divided by sin. Of 4 T. Now 4 T is the angle of the sign, and we also have it in the numerator, so it looks much better because it corresponds to the original identity of X divided by sign of X, right? But now we have changed 2T into 4T, right? So if we want to go back into the original 2T we are simply adding one half in the numerator, right? Because one half multiplied by 4T gives us 2 T. So we haven't really changed anything, right? We have simply arrived at 4T because this is what we desire to have in the numerator, and now from here we can apply the properties of limits. What we can do is simply factor out the constant negative 1/2 in front of the limit using the properties of limits, which gives us -12 multiplied by limit. As T approaches 0 off. 4 T divided by sin of 4 T. And then we have to understand that if T approaches 0, then multiplying both sides by 4, we also get 4 T approaches 0. Meaning we can say that. We get 1/2 multiplied by a limit as 4 T approaches 0 of 4T divided by sin of 4T which perfectly matches our original identity, right? We know that this limit. is equal to one. And therefore we can conclude that the answer is -12 multiplied by 1, which gives us -12 and therefore, the correct answer to this problem corresponds to the answer to C-12. Thank you for watching.

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limh→0− h / sin 3h

Key Concepts

Limit of a Function

Sine Function and Its Properties

L'Hôpital's Rule

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