L'Hopital's Rule

3 Key excerpts on "L'Hopital's Rule"

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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)

  x →∞. So L'Hospital's rule applies.

  This is still indeterminate as it stands, but it may be observed that if we apply L'Hospitals rule again, it will only make things worse. On the other hand, if we rewrite the bracketed expression as follows, the situation is simplified.

  Thus,

  L'Hospital's rule consists of several versions on the theme of using derivatives to evaluate limits of quotients. However, it is useful to have an overall view of the L'Hospital's rule, stated in simplified language in the next section.

  21.7 Most General Statement of L'Hospital's Theorem

  Theorem: Let f (x ) and g (x ) be two functions tending simultaneously to zero or infinity as x → u (or as x → ∞). If the ratio of their derivatives has a limit (finite or infinite) , the ratio of the functions possesses a limit that is equal to the limit of the ratio of the derivatives:

  Here, u may stand for a , a − , a + , −∞, or +∞.

  Note (15): (Historical Note) L'Hospital's rule should actually be called “Bernoulli's rule” because it appears in a correspondence from Johann Bernoulli to L'Hospital. L'Hospital and Bernoulli had made an agreement under which L'Hospital paid Bernoulli a monthly fee for solutions to certain problems, with the understanding that Bernoulli would tell no one of the arrangement. As a result, the rule described in the above theorem first appeared in L'Hospitals 1696 treatise. It was only recently discovered that the rule, its proof, and relevant examples all appeared in a 1694 letter from Bernoulli to L'Hospital.

  21.8 Meaning of Indeterminate Forms

  Certain limit problems have been classified as indeterminate forms . In fact, 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

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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)

  This result also applies when the limit of f (x)H20862g(x) as x approaches c produces any one of the indeterminate forms ∞ H20862 ∞ , (- ∞ )H20862 ∞ , ∞ H20862(- ∞ ), or (- ∞ )H20862(- ∞ ). A proof of this theorem is given in Appendix A. People occasionally use L’Hôpital’s Rule incorrectly by applying the Quotient Rule to f (x)H20862g(x). Be sure you see that the rule involves f uni2032 (x) guni2032 (x) not the derivative of f (x)H20862g(x). L’Hôpital’s Rule can also be applied to one-sided limits. For instance, if the limit of f (x)H20862g(x) as x approaches c from the right produces the indeterminate form 0H208620, then lim xuni2192c + f (x) g(x) = lim xuni2192c + f uni2032 (x) guni2032 (x) provided the limit exists (or is infinite). FOR FURTHER INFORMATION To enhance your understanding of the necessity of the restriction that guni2032 (x) be nonzero for all x in (a, b), except possibly at c, see the article “Counterexamples to L’Hôpital’s Rule” by R. P. Boas in The American Mathematical Monthly. To view this article, go to MathArticles.com. GUILLAUME L’HÔPITAL (1661–1704) L’Hôpital’s Rule is named after the French mathematician Guillaume François Antoine de L’Hôpital. L’Hôpital is credited with writing the first text on differential calculus (in 1696) in which the rule publicly appeared. It was recently discovered that the rule and its proof were written in a letter from John Bernoulli to L’Hôpital. “… I acknowledge that I owe very much to the bright minds of the Bernoulli brothers. … I have made free use of their discoveries …,” said L’Hôpital. See LarsonCalculus.com to read more of this biography. The Granger Collection, NYC 5.6 Indeterminate Forms and L’Hôpital’s Rule 347 Indeterminate Form 0H208620 Evaluate lim xuni21920 e 2x - 1 x . Solution Because direct substitution results in the indeterminate form 0H208620, lim xuni21920 (e 2x - 1) = 0 lim xuni21920 e 2x - 1 x lim xuni21920 x = 0 you can apply L’Hôpital’s Rule, as shown below.

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  eBook - ePub

  Mathematical Methods for Life Sciences

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

    (Publisher)

  Hence, we proved that f is continuous, but not differentiable, at the point x = 0. Looking at the above figure, this is geometrically clear: we cannot draw the tangent line at the point x = 0 to the graph of f. 2.7 De L’Hopital’s Rule The method of De L’Hopital (or De L’Hospital) allows us to compute limits leading to the indeterminate forms 0/0 or ∞/∞ through the derivative. We will state this result in the particular case, of great interest, of continuous, differentiable functions, with continuous derivative. We could prove this result also without such hypotheses, but since most functions we are interested in satisfy them, we will not examine the more general case. Theorem 2.7.1 De L’Hopital’s Rule: Let f, g : [ a, b ] → ℝ be continuous functions in [ a, b ], differentiable in (a, b) except, eventually, in c ∈ (a, b). Let g ′ (x) ≠ 0, for x ≠ c and suppose: lim x → c f (x) = lim x → c g (x) = 0 or lim x → c f (x) = ± ∞, lim x → c g (x) = ± ∞ Moreover. assume: lim x → c f ′ (x) g ′ (x) = L Then lim x → c f (x) g (x) = L (Notice L can also be ±∞). Proof. We prove the result only for the case c ∈ ℝ, that is, the values of the functions f, g approaching zero, as x approaches c. The general case is a variation of the argument given below. We have. that lim x → c f (x) g (x) = lim x → c f (x) − 0 g (x) − 0 = lim x → c f (x) − f (c) g (x) − g (c) = lim x → c (f (x) − f (c) x − c) (g (x) − g (c) x[-. -=PLGO-SEPARATOR=--]− c) = lim x → c (f (x) − f (c) x − c) lim x → c (g (x) − g (c) x − c) = f ′ (x) g ′ (x) Observation 2.7.2 At this point, the student may be tempted to prove the standard limits in Proposition 2.2.4 using De L’Hospital’s rule, but this would be a serious logical error. Let us see why. Suppose we want to prove the standard limit: lim x → 0 e x − 1 x using De L’Hospital’s rule

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