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APPLICATION TO THE EVALUATION OF

SINGULAR FORMS AND MAXIMA

AND MINIMA VALUES.

CHAPTER XIII.

UNDETERMINED FORMS.

327. In Chap. I. it was explained that a function may involve an independent variable in such a manner that its value for a certain assigned value of the variable cannot be found by a direct substitution of that value. And in such cases the function is said to assume a "Singular," "Undetermined," "Illusory," or "Indeterminate form.

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328. It is proposed in the present chapter to consider more fully the method of evaluation of the true limiting values of such quantities when the independent variable is made to approach indefinitely near its assigned value.

329. List of Forms occurring.

Several cases are to be considered, viz., when, upon substitution of the assigned value of the independent variable, the function reduces to one of the forms

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It is frequently easy to treat these cases by algebraical

or trigonometrical methods without having recourse to the Differential Calculus, though the latter is required for a general discussion of such forms.

By far the most important case to consider is that in

which the function takes the form

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; for, in the first

place, it is the one which most frequently occurs; and, secondly, any of the other forms may be made to depend upon this one by some special artifice.

330. Algebraical Treatment.

Suppose the function to take the form when the independent variable x ultimately coincides with its assigned value a. Put x=a+h and expand both numerator and denominator of the function. It will now become apparent that the reason why both numerator and denominator vanish is that some power of h is a common factor of each. This should now be divided out. Finally, put h=0 so that x becomes = a, and the true limiting value of the function will be apparent.

In the particular case in which x is to become zero the expansion of numerator and denominator in powers of a should be at once proceeded with without any preliminary substitution for x.

In the case in which x is to become infinite, put x=

so that when x becomes = ∞ y becomes =0.

1

y'

The method thus explained will be better understood by examining the mode of solution of the following examples.

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