Schaum's outline of theory and problems of tensor calculus by Kay D.

By Kay D.

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EXERCISE 1—3 Evaluate each of the following limits: The Derivative of a Function CHAPTER TWO INCREMENT NOTATION 2—1. A Convenient Symbolism. ” We shall now introduce a convenient notation for small intervals, or for small changes in the value of a variable or a function. When a variable changes from one numerical value to another, we refer to the difference of the values as the increment of the variable; for example, if x changes from x1 to x2, then the increment equals x2 − x1. The increment is represented by the symbol Δx, read “delta x”; thus Δx = x2 − x1.

Substituting 0 for x yields , which is indeterminate. Hence, divide both numerator and denominator by x2, the lowest power of x in the function: EXAMPLE 11. Divide both numerator and denominator by x3, the highest power of x in the function. EXAMPLE 12. Solution. Hence EXAMPLE 13. Substituting x = 3 yields = which is indeterminate. By factoring: 1—17. A More Precise Definition of a Limit. 002 inch. Call the measured dimension of the finished piece D′ and the difference D′ − D = E. ) By using appropriate tools and skill, he can make the value of E as close to zero as he wishes.

If we take values of x successively nearer to 3, so that |x − 3| becomes less and less, the quantity |f(x) − 19| will also become less and less. Now let us select any number, as small as we please, say . It is then possible to choose |x − 3| so as to make |f(x) − 19| less than . ” The limit in this illustration is exactly 19. The limit of a function is the exact value toward which the function approaches as x comes sufficiently near to some fixed value. The value of the function may or may not reach the limit toward which it approaches.

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