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Quite often, functions are not defined in certain points, e.g. if the used functional term is not defined there. However, it makes a huge difference whether only the functional term is not defined in a point, but has a useful (continuous) extension, or whether the function does not have a useful extension in this point, e.g. because it has a pole or an even more chaotic behavior. The following concept is in particular relevant for the definition of differentiability (if the difference quotient has a useful limit, then it is called differential quotient).

Let denote a subset and a point. Let

be a function. Then is called limit of in , if for every there exists some such that for all fulfilling

the estimate

holds. In this case, we write

This concept is basically only useful if there exists at least some sequence within converging to . A typical situation is the following: Let denote a real interval, a point and let . The function is defined on but not in , and we are dealing with the question whether can be extended to a function defined on the whole . Here, should be determined by .

Let denote a subset and a point. Let

be a function and

a point. Then the following statements are equivalent.
  1. We have
  2. For every sequence in which converges to , also the image sequence converges to .


This implies for a continuous function that it can be extended to a continuous function (by ) if and only if the limit of in equals .

Let denote a subset and a point. Let and denote functions, such that the limits and

exist. Then the following statements hold.
  1. The sum has in the limit
  2. The product has in the limit
  3. Suppose that for all and . Then the quotient has in the limit

This follows from fact and from fact.

We consider the limit

where . For , this term is not defined, and from this term one can not read of directly whether the limit exists. It is however possible to multiply the numerator and the denominator by , then we get

Due to the rules for limits, we can determine the limit in the numerator and in the denominator separately, where for the denominator we use the continuity of the square root according to exercise. Hence, the limit is .