Find the limits in Exercises 31–40. Are the functions continuo...

Question

Find the limits in Exercises 31–40. Are the functions continuous at the point being approached?

lim t → 0 sin (π/2 cos (tan t))

Accepted Answer

Welcome back, everyone. In this problem, we want to find the limit as X approaches a 1/2 of i of the expression, the cosine of 14th of pi multiplied by the sine of the cosine of X. Determine if the function is continuous at the point being approached. Now if we're going to find the limit of this expression as X approaches a half of pi, we need to evaluate our inner functions. Notice that our X term is in the innermost function, the cosine of X. So let's evaluate the cosine of X as X approaches a half of pi and then continue to gradually evaluate the rest of the innermost function, which will eventually help us find the limit. So in other words, we are solving for the limit. As x approaches 1/2 of pi, of the cosine of 4th of pi multiplied by the sine of a cosine of X. Now, as X, as you say, we're starting with the innermost functions, as X approaches a half of pi, then the cosine of a half of, sorry, the cosine of X, OK. The coign of X. Is going to approach, OK, the cosign of a half of pi, which is going to be 0. So a cosine, as X approaches a 1/2 of pi, the cosine of X will approach 0 because the cosine of a 1/2 of pi equals 0. Thus, OK, that means then that the sign of the cosine of X is also going to approach the sign of 0, which equals 0. And that means. That 1/4 of pi multiplied by the sine of the cosine of X. Is also going to approach a 4th of pi multiplied by 0, which equals 0. So now to compute the limit, then that means that the limit as X approaches a half of pi. Of the cosine of a 4th of pi multiplied by the sine of the cosine of X. Is going to approach the cosine of 0, which is equal to 1, thus the limit is equal to 1. Now that we have the limit, we need to determine if the function is continuous at the point being approached. How can we do that? Well, to check for continuity. OK, to check for continuity. If we say that F of X is equal to the function, the cosine of 1/4 of pi multiplied by the sign of the cosine of X. Then we know that it's continuous if the limit as x approaches a half of of F of X. Is equal to 5/15 of 5. No, we know that this limit is 1. So now, we need to test. If we put a half of pi into our function, will it also be equal to 1? Well, F of 1/2 of pi, OK, is gonna be equal to the cosine of 4th of pi multiplied by the sin. Of the cosign of a half of pie, OK? We know that a cosign of a half of pie. Cos for half of pie is equal to. 0, sorry, yes, the cosine of half of pi, sorry, is equal to 0. So if we go ahead and substitute here, then this is gonna be equal to the cosine of 0, which equals 1. Thus, that is how we know that our limit as X approaches a half of pi of F of X is equal to F1/2 of pi, which means then that the function is continuous. Because it matches the limit. Thus, the function's limit is 1, and it is continuous at that point. Thanks a lot for watching everyone. I hope this video helped.

Find the limits in Exercises 31–40. Are the functions continuous at the point being approached?

lim t → 0 sin (π/2 cos (tan t))

Key Concepts

Limits

Continuity

Trigonometric Functions

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