Here we show some examples on separating "nice" parts from "tough" parts, represented by a formula f (n).

Note that in the first two examples we eventually dropped the first limit and left just the second to evaluate, while in the third example we kept the zero. Here a beginner may be even excused for not evaluating the second limit at all and proclaim that the whole problem has answer 0. However, we know that the theorem on limits and operations can be used only if the outcome of the operations makes sense; that is, if we do not get an indeterminate expression. In the first two examples, no matter what the second limit is, we can always get an answer to the whole problem using limit algebra, since there is no indeterminate expression of the form "0 + (  )" or "1⋅(  )". However, in the third example we do not know. If the second limit did not exist or gave infinity, we could not determine the outcome and therefore the separation into two limits could not be done. We would have to go back to the beginning and try some tricks. Thus here one has to finish the calculation to see what will come of "0⋅(  )".