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what's the limit of 3^x +2/e^2x +1 as x goes to infinity

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you can throw away the +2 and +1, since they become insignificant.
Then you are left with just 3^x/e^(2x) = 3^x/(e^2)^x = (3/e^2)^x
since 3/e^2 < 1, the limit is zero

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