Use continuity to evaluate the limit. lim x→π 5 sin(x + sin x)

You can stop right there, or you can try finding the exact value of .

Recall DeMoivre's theorem:



This means when , the imaginary part of the expansion of the left side will give you an expanded form of in terms of powers of . You have




where the last equality comes from the fact that . So



Now, setting , you get




Clearly, , so you're left with the quartic equation



Applying the quadratic formula gives a solution of



Since , we should expect to be smaller, which means we take the positive root because , and adding a positive number would make this larger. So,



which means



but we also expect this number to be positive, so we ignore the negative root and end up with



So the limit is



Now, there's no reason to expect this to have a simpler form, so we can stop here. (Perhaps this answer is overkill, but if you didn't know this stuff, it doesn't hurt to learn it.)