4.6. Indeterminate Forms and L Hôpital s Rule. 292 Chapter 4: Applications of Derivatives. Indeterminate Form 0/0

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1 292 Chapter 4: Applications of Derivatives 4.6 Indeterminate Forms and L Hôpital s Rule HISTORICAL BIOGRAPHY Guillaume François Antoine de l Hôpital (66 74) John Bernoulli discovered a rule for calculating its of fractions whose numerators and denominators both approach zero or + q. The rule is known today as l Hôpital s Rule, after Guillaume de l Hôpital. He was a French nobleman who wrote the first introductory differential calculus text, where the rule first appeared in print. Indeterminate Form / If the continuous functions ƒ(x) and g(x) are both zero at x = a, then gsxd cannot be found by substituting x = a. The substitution produces >, a meaningless expression, which we cannot evaluate. We use > as a notation for an expression known as an indeterminate form. Sometimes, but not always, its that lead to indeterminate forms may be found by cancellation, rearrangement of terms, or other algebraic manipulations. This was our experience in Chapter 2. It took considerable analysis in Section 2.4 to find x: sd>x. But we have had success with the it - ƒsad ƒ sad =, from which we calculate derivatives and which always produces the equivalent of > when we substitute x = a. L Hôpital s Rule enables us to draw on our success with derivatives to evaluate its that otherwise lead to indeterminate forms. THEOREM 6 L Hôpital s Rule (First Form) Suppose that ƒsad = gsad =, that ƒ sad and g sad exist, and that g sad Z. Then gsxd = ƒ sad g sad.

2 4.6 Indeterminate Forms and L Hôpital s Rule 293 Caution To apply l Hôpital s Rule to ƒ>g, divide the derivative of ƒ by the derivative of g. Do not fall into the trap of taking the derivative of ƒ>g. The quotient to use is ƒ >g, not sƒ>gd. Proof Working backward from ƒ sad and g sad, which are themselves its, we have ƒ sad g sad = - ƒsad gsxd - gsad = = - ƒsad gsxd - gsad = - ƒsad gsxd - gsad - gsxd - = gsxd. EXAMPLE 3x - x: x 2 + x - x: x = Using L Hôpital s Rule = 3 - cos x ` = 2 x= 22 + x 3 x = = 2 Sometimes after differentiation, the new numerator and denominator both equal zero at x = a, as we see in Example 2. In these cases, we apply a stronger form of l Hôpital s Rule. THEOREM 7 L Hôpital s Rule (Stronger Form) Suppose that ƒsad = gsad =, that ƒ and g are differentiable on an open interval I containing a, and that g sxd Z on I if x Z a. Then gsxd = assuming that the it on the right side exists. ƒ sxd g sxd, Before we give a proof of Theorem 7, let s consider an example. EXAMPLE 2 Applying the Stronger Form of L Hôpital s Rule x: 2 + x - - x>2 x 2 x - x: x 3 = s>2ds + xd ->2 - >2 x: 2x = -s>4ds + xd -3>2 =- x: 2 8 = x: - cos x 3x 2 = x: 6x = cos x x: 6 = 6 Still differentiate again. ; Not it is found. ; Still Still Not it is found. ;

3 294 Chapter 4: Applications of Derivatives The proof of the stronger form of l Hôpital s Rule is based on Cauchy s Mean Value Theorem, a Mean Value Theorem that involves two functions instead of one. We prove Cauchy s Theorem first and then show how it leads to l Hôpital s Rule. HISTORICAL BIOGRAPHY Augustin-Louis Cauchy ( ) THEOREM 8 Cauchy s Mean Value Theorem Suppose functions ƒ and g are continuous on [a, b] and differentiable throughout (a, b) and also suppose g sxd Z throughout (a, b). Then there exists a number c in (a, b) at which g scd = gsbd - gsad. y (g(c), f(c)) (g, f) slope (g, f) f f g g FIGURE 4.42 There is at least one value of the parameter t = c, a 6 c 6 b, for which the slope of the tangent to the curve at (g(c), ƒ(c)) is the same as the slope of the secant line joining the points (g, ƒ) and (g, ƒ). x Proof We apply the Mean Value Theorem of Section 4.2 twice. First we use it to show that gsad Z gsbd. For if g did equal g, then the Mean Value Theorem would give for some c between a and b, which cannot happen because g sxd Z in (a, b). We next apply the Mean Value Theorem to the function This function is continuous and differentiable where ƒ and g are, and Fsbd = Fsad =. Therefore, there is a number c between a and b for which F scd =. When expressed in terms of ƒ and g, this equation becomes or g scd = Fsxd = - ƒsad - F scd = - g scd gsbd - gsad b - a [ gsxd - gsad]. gsbd - gsad gsbd - gsad [ g scd] = = gsbd - gsad. Notice that the Mean Value Theorem in Section 4.2 is Theorem 8 with gsxd = x. Cauchy s Mean Value Theorem has a geometric interpretation for a curve C defined by the parametric equations x = gstd and y = ƒstd. From Equation (2) in Section 3.5, the slope of the parametric curve at t is given by dy>dt dx>dt = ƒ std g std, so >g scd is the slope of the tangent to the curve when t = c. The secant line joining the two points (g, ƒ) and (g, ƒ) on C has slope gsbd - gsad. Theorem 8 says that there is a parameter value c in the interval (a, b) for which the slope of the tangent to the curve at the point (g(c), ƒ(c)) is the same as the slope of the secant line joining the points (g, ƒ) and (g, ƒ). This geometric result is shown in Figure Note that more than one such value c of the parameter may exist. We now prove Theorem 7. =

4 4.6 Indeterminate Forms and L Hôpital s Rule 295 Proof of the Stronger Form of l Hôpital s Rule We first establish the it equation for the case x : a +. The method needs almost no change to apply to x : a -, and the combination of these two cases establishes the result. Suppose that x lies to the right of a. Then g sxd Z, and we can apply Cauchy s Mean Value Theorem to the closed interval from a to x. This step produces a number c between a and x such that But ƒsad = gsad =, so g scd As x approaches a, c approaches a because it always lies between a and x. Therefore, + gsxd = = - ƒsad gsxd - gsad. g scd = gsxd. c:a + g scd = ƒ sxd + g sxd, which establishes l Hôpital s Rule for the case where x approaches a from above. The case where x approaches a from below is proved by applying Cauchy s Mean Value Theorem to the closed interval [x, a], x 6 a. Most functions encountered in the real world and most functions in this book satisfy the conditions of l Hôpital s Rule. Using L Hôpital s Rule To find gsxd by l Hôpital s Rule, continue to differentiate ƒ and g, so long as we still get the form > at x = a. But as soon as one or the other of these derivatives is different from zero at x = a we stop differentiating. L Hôpital s Rule does not apply when either the numerator or denominator has a finite nonzero it. EXAMPLE 3 Incorrectly Applying the Stronger Form of L Hôpital s Rule Not it is found. ; Up to now the calculation is correct, but if we continue to differentiate in an attempt to apply l Hôpital s Rule once more, we get - cos x x: x + x 2 x: - cos x x + x 2 = x: + 2x = = = x: + 2x = cos x x: 2 = 2, which is wrong. L Hôpital s Rule can only be applied to its which give indeterminate forms, and > is not an indeterminate form.

5 296 Chapter 4: Applications of Derivatives L Hôpital s Rule applies to one-sided its as well, which is apparent from the proof of Theorem 7. EXAMPLE 4 Using L Hôpital s Rule with One-Sided Limits Recall that q and + q mean the same thing. x: + x 2 x: - x 2 cos x = x: + 2x cos x = x: - 2x = q = -q Positive for x 7. Negative for x 6. Indeterminate Forms ˆ> ˆ, ˆ #, ˆ ˆ Sometimes when we try to evaluate a it as x : a by substituting x = a we get an ambiguous expression like q> q, q #, or q - q, instead of >. We first consider the form q> q. In more advanced books it is proved that l Hôpital s Rule applies to the indeterminate form q> q as well as to >. If : ; q and gsxd : ; q as x : a, then gsxd = ƒ sxd g sxd provided the it on the right exists. In the notation x : a, a may be either finite or infinite. Moreover x : a may be replaced by the one-sided its x : a + or x : a -. EXAMPLE 5 Find x:p>2 sec x + tan x x - 2x 2 3x 2 + 5x Working with the Indeterminate Form q> q Solution The numerator and denominator are discontinuous at x = p>2, so we investigate the one-sided its there. To apply l Hôpital s Rule, we can choose I to be any open interval with x = p>2 as an endpoint. x:sp>2d - from the left The right-hand it is also, with s - q d>s - q d as the indeterminate form. Therefore, the two-sided it is equal to. sec x + tan x x - 2x 2 3x 2 + 5x = = x:sp>2d - sec x tan x sec 2 x q q - 4x 6x + 5 = = x:sp>2d - = -4 6 =-2 3.

6 4.6 Indeterminate Forms and L Hôpital s Rule 297 Next we turn our attention to the indeterminate forms q # and q - q. Sometimes these forms can be handled by using algebra to convert them to a > or q> q form. Here again we do not mean to suggest that q # or q - q is a number. They are only notations for functional behaviors when considering its. Here are examples of how we might work with these indeterminate forms. EXAMPLE 6 Find Working with the Indeterminate Form ax sin x b q # Solution ax sin x b q # = a sin hb h: + h = Let h = >x. EXAMPLE 7 Find Working with the Indeterminate Form a x: - x b. q - q Solution If x : +, then : + and Similarly, if x : -, then : - and - x : q - q. - x : - q - s - q d = -q + q. Neither form reveals what happens in the it. To find out, we first combine the fractions: - x = x - x Common denominator is x Then apply l Hôpital s Rule to the result: a x: - x b = x: x - x = x: - cos x + x cos x = x: 2 cos x - x = 2 =. Still