Indeterminate Forms

4 Key excerpts on "Indeterminate Forms"

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  eBook - ePub

  Introduction to Differential Calculus

  Systematic Studies with Engineering Applications for Beginners

  • Ulrich L. Rohde, G. C. Jain, Ajay K. Poddar, A. K. Ghosh(Authors)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  the term indeterminate form is used to say that the result is not obvious. We classify them as follows:

  i. Indeterminate Limit Problems of the Form and ±

  (Quotient Forms): Consider the Limits,

  These examples show that one could define to be 0, 2, or ∞ with equal justification . It is for this reason that one does not attempt to define . This expression is an example of an indeterminate form (see Chapter 1).

  Next, consider the limit problems in the form :

  where both numerator and denominator in each limit approach ∞ as x approaches 2. These examples suggest that we should consider to be an indeterminate form.

  Note that is not an indeterminate form , for if , then

  Also, is not an indeterminate form, for if , then is always undefined, and the quotient becomes large in absolute value as x approaches a. Thus, and , are both not Indeterminate Forms.

  ii. Indeterminate Limit Problems in Product Forms [0 · or · 0]: Consider the limits and .

  These examples show that we should consider 0

  · ∞ (or · 0) to be an indeterminate form.

  Note that the product (∞ · ∞) is not an indeterminate product form (why ?).

  iii. Indeterminate Sum and Difference [(− ) + (or − )]: Consider the limits

  and These examples show that −(∞)+∞ and ∞ − ∞ should be considered to be of indeterminate form.

  Note that, the sum ∞ + ∞ is not an indeterminate form (why?).

  iv. The Indeterminate Exponential Forms [0 · 0, 1 · ∞, 1−∞, ∞ · 0]: Indeterminate exponential forms arise from expressions of the type

  Recall that we have defined the exponential r s for all s only r > 0.

  Hence, we assume that f (x ) > 0 for x ≠ a .

  Since the logarithm function is continuous and is the inverse of the exponential function, we see that,

  so that we can write

  (Recall that ln x (= loge x ) = b means eb = x

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  eBook - PDF

  Calculus of a Single Variable: Early Transcendental Functions, International Metric Edition

  • Ron Larson, Bruce Edwards(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Apply L’Hôpital’s Rule to evaluate a limit. Indeterminate Forms Recall from Chapters 2 and 4 that the forms 0H208620 and ∞ H20862 ∞ are called indeterminate because they do not guarantee that a limit exists, nor do they indicate what the limit is, if one does exist. When you encountered one of these Indeterminate Forms earlier in the text, you attempted to rewrite the expression by using various algebraic techniques. Indeterminate Form Limit Algebraic Technique 0 0 lim xuni2192-1 2x 2 - 2 x + 1 = lim xuni2192-1 2(x - 1) Divide numerator and denominator by (x + 1). = -4 ∞ ∞ lim xuni2192 ∞ 3x 2 - 1 2x 2 + 1 = lim xuni2192 ∞ 3 - (1H20862x 2 ) 2 + (1H20862x 2 ) Divide numerator and denominator by x 2 . = 3 2 Occasionally, you can extend these algebraic techniques to find limits of transcendental functions. For instance, the limit lim xuni21920 e 2x - 1 e x - 1 produces the indeterminate form 0H208620. Factoring and then dividing produces lim xuni21920 e 2x - 1 e x - 1 = lim xuni21920 (e x + 1)(e x - 1) e x - 1 = lim xuni21920 (e x + 1) = 2. Not all Indeterminate Forms, however, can be evaluated by algebraic manipulation. This is often true when both algebraic and transcendental functions are involved. For instance, the limit lim xuni21920 e 2x - 1 x produces the indeterminate form 0H208620. Rewriting the expression to obtain lim xuni21920 parenleft.alt4 e 2x x - 1 x parenright.alt4 merely produces another indeterminate form, ∞ - ∞ . Of course, you could use technology to estimate the limit, as shown in the table and in Figure 5.42. From the table and the graph, the limit appears to be 2. (This limit will be verified in Example 1.) x -1 -0.1 -0.01 -0.001 0 0.001 0.01 0.1 1 e 2x - 1 x 0.865 1.813 1.980 1.998 ? 2.002 2.020 2.214 6.389 x e 2x - 1 x y = y - 1 - 2 - 3 - 4 1 2 3 4 2 3 4 5 6 7 8 The limit as x approaches 0 appears to be 2. Figure 5.42

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  eBook - ePub

  Mathematical Methods for Life Sciences

  • Cinzia Bisi, Rita Fioresi(Authors)
  • 2024(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  We invite the student to make use of a graphic software to check geometrically all these inequalities. We notice also, by looking at the picture that n and m may also take rational values and all the statements are still true. Of course this is not a mathematical proof, but it gives a geometric intuition to comprehend our statements. The following proposition is very important to compute limits involving Indeterminate Forms. Proposition 2.3.5 Let f, g : D → ℝ be two functions, with f, g infinite for x approaching c. If ord ∞ (f) > ord ∞ (g), then: lim x → c f (x) + g (x) = lim x → c f (x) In other words: in a sum of infinites, we can ignore the lower-order infinite functions. Proof. We can. write: lim x → c f (x) + g (x) = lim x → c f (x) [ 1 + g (x) / f (x) ] = lim x → c f (x) Now we see some examples of applications of the notion of order of infinity. Example 2.3.6 Consider lim x →−∞ x 3 − 4 x = −∞ + ∞. By the previous proposition, with f (x) = x 3 and g (x) = −4 x, we can ignore g (x), hence: lim x → − ∞ x 3 − 4 x = lim x → − ∞ x 3 = − ∞ Consider lim x → − ∞ x 3 − 4 x x 4 − 3 x + 2 = − ∞ / ∞. By the previous proposition, at the numerator we can ignore −4 x, and at the denominator we can ignore −3 x + 2, because they are infinites of order less than x 3 e x 4 respectively. Hence: lim x → − ∞ x 3 − 4 x x 4 − 3 x + 2 = lim x → − ∞ x 3 x 4 = lim x → − ∞ 1 x = 0 Consider the limit: lim x → + ∞ e x + x + log (x) 2 + x 3 − log (x) = ∞ ∞ − ∞ At the numerator, since e x has higher order of infinity, we can ignore the other terms. In the denominator, the higher order infinite is x 3, so we can ignore the other terms. lim x → + ∞ e x + x + log (x) 2 + x 3 − log (x) = lim x → + ∞ e x x 3 = + ∞ The last equality is due again to the fact that e x has higher order of infinity. We now proceed to describe a method for solving Indeterminate Forms of the type 0 0. Definition 2.3.7 Let us consider a function f : D → ℝ

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  The philosophy of mathematics

  • Auguste Comte(Author)
  • 2012(Publication Date)
  • Perlego

    (Publisher)

  Method of indeterminate Coefficients. This method, so eminently analytical, and which must be regarded as one of the most remarkable discoveries of Descartes, has undoubtedly lost some of its importance since the invention and the development of the infinitesimal calculus, the place of which it might so happily take in some particular respects. But the increasing extension of the transcendental analysis, although it has rendered this method much less necessary, has, on the other hand, multiplied its applications and enlarged its resources; so that by the useful combination between the two theories, which has finally been effected, the use of the method of indeterminate coefficients has become at present much more extensive than it was even before the formation of the calculus of indirect functions.

  Having thus sketched the general outlines of algebra proper, I have now to offer some considerations on several leading points in the calculus of direct functions, our ideas of which may be advantageously made more clear by a philosophical examination.

  IMAGINARY QUANTITIES.

  The difficulties connected with several peculiar symbols to which algebraic calculations sometimes lead, and especially to the expressions called imaginary, have been, I think, much exaggerated through purely metaphysical considerations, which have been forced upon them, in the place of regarding these abnormal results in their true point of view as simple analytical facts. Viewing them thus, we readily see that, since the spirit of mathematical analysis consists in considering magnitudes in reference to their relations only, and without any regard to their determinate value, analysts are obliged to admit indifferently every kind of expression which can be engendered by algebraic combinations. The interdiction of even one expression because of its apparent singularity would destroy the generality of their conceptions. The common embarrassment on this subject seems to me to proceed essentially from an unconscious confusion between the idea of function and the idea of value, or, what comes to the same thing, between the algebraic and the arithmetical

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