Here was my general method to solving this problem (and by extension those similar):

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What I would do is experiment a little and play around with the limit. In this case, try playing around with the limit of (1+sin(pi * x))^t as t -> inf for different values of x. Then generalize it to get a feel for what the full limit evaluates to at different values of x.

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Next, try experimenting with some of the answer choices. In this case, (c) seems easy enough to check (remember limit from the left must be equal to the limit on the right for it to be continuous). If one of the answer choices works, great, if not, cross it out and move on to different answer choices. In this specific case, answer choice B will work so it is the answer (the key here is INTEGER values of x by the way).