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Problem 28 Easy Difficulty

Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain.

$ R(t) = \dfrac{e^{\sin t}}{2 + \cos \pi t} $

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Video Transcript

this problem Number twenty eight of the Stewart character Safe Addition Section two point five explained using therms four, five, seven and nine Why the function is continuous that every number in its domain state to domain the function Our of team is equal to deed to the sign of team divided by the quantity two plus cosign of high team. And if we were caller theorems, this would fall under Dan for where we have ah quotient a ratio of two functions. It would belong to the exponential functions on exponential function of which is Russia. And it is continuous on it so mean and in the denominator we have a trick in magic function which is also included and being continuous on its domain. Ah, the only concern would be that for this ratio of functions that the denominator not be zero. Make sure that we don't include that those numbers in the domain. And if we were to ask ourselves what values for teen would make this their own and we attempt to solve for these eyes, we get this statement What is cosign a pie I'm seeing. When is that equal to negative too? But co sign has a range as a range of values when plotted on ly between positive one and negative one. So it only goes his highest positive one When they go low is negative one and so it would never be able to reach native to Therefore this down later can never be equal to zero And overall this Then I'm near we'LL never be negative as well But the important part is that there are no points to restrict the domain for this problem It's going to be all rials all real numbers and we can definitively state that this function is continuous understanding.