Box "a nice outer function"

Often one needs to find the limit at a of an expression that has the form of a composed function f (g). If f is "nice" (for instance, if it is continuous on a neighborhood of g(a)), then we can "pull f out of the limit", that is, we find the limit of g at a and then put this outcome into f. We can also write it like this:

This is often used to simplify calculations, for instance here it makes the difference between doing l'Hospital's rule once or 13 times. Sometimes we cannot use suitable methods without this rule, for instance it is one of the basic steps when dealing with indeterminate powers.

Another example can be found in Solved Problems - Limits.

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