By Smirnov V.I.

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Extra resources for A course of higher mathematics, vol. 2

Example text

Indeed, we saw that the values of f (x) can get arbitrarily close to 1 if the real numbers x are chosen from a suitably small interval around 0. At this point, one could ask the following question. 0001 not? 0001. In order to answer this question, we must have a good understanding of the definition of limits. That definition says that if limx→0 f (x) = L, then the values of f (x) will get arbitrarily close to f (0) if x is chosen from a suitably small interval around 0. The key word in the previous sentence is arbitrarily.

2. The Derivative of the Function f . Recall that, at a given point a, the derivative of the function f is defined as the limit f (x) − f (a) . f (a) = lim x→a x−a Note that this definition associates the real number f (a) to the real number a. That is, f : R → R is a function. The function f is called the derivative of f . The operation that takes f into f is called differentiation. This explains the following definition. 20. A function f is called differentiable at a if f (a) exists. We say that f is differentiable on the interval (a, b) if f is differentiable at d for all d ∈ (a, b).

Then the limit of g at a = 0 does not exist. Indeed, no matter how small an interval I we take around the point a = 0, that interval I will contain some positive and some negative real numbers. Hence, the values of g(x) will sometimes equal 1 and sometimes equal 0 for x ∈ I, no matter how small I is. 5 to it. So limx→0 g(x) does not exist. Second, if limx→0 f (x) exists, it is unique; that is, f cannot have two different limits at any given point a. Let us illustrate this using the introductory example of this section, the function f (x) = 1/(1+x).

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