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 x → 0 tan (π/4 cos (sin x¹/³))

Accepted Answer

Welcome back, everyone. In this problem, we want to find the limit as X approaches 0 of the expression, the tangent of a 6th of pi multiplied by the sign of a 1/2 of pi of the cosine of the X-rays to 1/5. Then we want to determine if the function is continuous at the point being approached. Now, for us to determine the limit, let's start by evaluating the inner functions, that is where the X term is, the cosine of X-rays to the 5th, and then after evaluating our innermost, we continue to evaluate outward until we can eventually find the limit as X approaches 0. Now, starting with the innermost function, OK. As X approaches 0, then the cosine of X, OK, approaches the cosine, sorry, the cosine of X to the 5th, OK. Is going to approach the cosine of 0 to 5th. Which equals the cosine of 0, which equals 1. Thus This tells us then, OK, that our function, the sign of a half of pi multiplied by the cosine of X to a 5th, OK. Is going to be equal. The design of a half of pi multiplied by 1. Which is gonna be equal to the size of a half of pi, which equals 1. If we take it a step further, not to go to our tangent, then this means that our tangent of a 6th of pi can multiplied by everything that was in here which now we know equals 1 is going to be the tangent of a 16th of pi multiplied by 1 which equals the tangent of a 16th of pi, which we know is equal to root 3 divided by 3. Thus this tells us then. That our limit as X approaches 0 of our expression, the tangent of a 6th of pi multiplied by the sign of a 1/2 of of a cosine of X to the 5th, OK. So remember how many brackets that we have. Is going to be equal to the tangent of a 6th of pi which we just found to be root 3 divided by 3. Thus, this is the limit of our function. Now let's look at the second part of our problem. How can we tell if the function is continuous at the point? Well, to check for continuity, the function, OK, and let's call all of this the tangent of our entire thing here or functions. So let's call all of this F of X, OK. Then the function F of X, let me just make some space here. Is going to be continuous, OK. If the limit as X approaches 0 of FFX. Is equal to f of 0. In other words, if we substitute 0 into our function and we get our limit, then we know that our function is continuous at that point. So that means then? That let me come down here. If we substitute X equals 0 directly into our function, F of 0 would be equal to the tangent of a 16th of pi multiplied by the sign of a 1/2 of pi of the cosine of X rays to a 5th, OK. And we know, let me make sure I have all my brackets here, we know that The cosine of 0 to a 5th is the cosine of 0, which would be equal to, as we found earlier one. So if we substitute 0 into our function, then we're still going to get route 3 divided by 3. I know since the functions limit matches its value at X equals 0, that means the function F of X is continuous. At that point. Thus, the limit of our function is Route 3 divided by 3, and our function 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 x → 0 tan (π/4 cos (sin x¹/³))

Key Concepts

Limits

Continuity

Trigonometric Functions

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