Bloomsburg University Bloomsburg, Pennsylvania 17815

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1 Department of Mathematics, Computer Science, and Statistics Bloomsburg University Bloomsburg, Pennsylvania L Hôpital s Rule Summary Many its may be determined by direct substitution, using a geometric argument, or some algebraic technique. However, various indeterminate forms, particularly those involving both algebraic and transcendental functions, require L Hôpital s Rule. This presentation will focus on motivating and teaching L Hôpital s Rule, AP type problems involving several indeterminate forms, numerical and graphical confirmation, and some interesting applications. Address: Stephen Kokoska Department of Mathematics, Computer Science, and Statistics Bloomsburg University 400 East Second Street Bloomsburg, PA Phone: (570) skokoska@planetx.bloomu.edu

2 Background: 1. When solving previous it problems we saw the indeterminate forms 0/0 and /. (There are other indeterminate forms.) 2. Various algebraic techniques were used. Example: x 1 x 2 x x 2 1 = x 3 1 Example: x 5x 3 2x + 7 = 3. Consider a procedure to solve problems like: (a) x 0 2 x 1 x ln x (b) x 1 x 1 (c) ln x x x 4. To solve these problems we use L Hôpital s Rule.

3 L Hôpital s Rule Suppose f and g are differentiable and g (x) 0 near a (except possibly at a). Suppose that f(x) = 0 and g(x) = 0 x a x a or that f(x) = ± and g(x) = ± x a x a (In other words, we have an indeterminate form of type 0/0 or /.) Then f(x) x a g(x) = f (x) x a g (x) if the it on the right side exists (or is or ). Note: 1. In words: The it of a quotient of functions is equal to the it of the quotient of their derivatives, provided the conditions are satisfied. It is important to verify the conditions regarding the its of f and g before using L Hôpital s Rule. 2. L Hôpital s Rule is also valid for one-sided its and for its at infinity or negative infinity. That is, x a can be replaced by any of the following symbols. x a +, x a, x, x. 3. For the special case in which f(a) = g(a) = 0, f and g are continuous, and g (a) 0, consider the following proof. f(x) f(a) f (x) x a g (x) = f (a) g (a) = x a x a g(x) g(a) x a x a = x a f(x) f(a) x a g(x) g(a) x a = x a f(x) f(a) g(x) g(a) = x a f(x) g(x)

4 4. Some graphical evidence: (a) The first graph shows two differentiable functions f and g. Each approaches 0 as x a. (b) If we zoom in around (a, 0), the graphs become locally linear. If the functions were actually linear, then their ratio would be which is the ratio of their derivatives. m 1 (x a) m 2 (x a) = m 1 m 2 This suggests that f(x) x a g(x) = f (x) x a g (x) y y = f(x) y y = m 1 (x a) PSfrag replacements PSfrag replacements a y = g(x) x a x y = m 2 (x a) 5. The first appearance in print of l Hôpital s Rule was in the book Analyse des Infiniment Petits published by the Marquis de l Hôpital in The example used to illustrate the rule was x a 2a 3 x x 4 a 3 a 2 x a 4 ax 3 where a > 0.

5 ln x Example: x 1 x 1 Example: x 0 cos x 1 x 2 Example: x π sin x 1 cos x

6 Example: x 0 2 x 1 x x Example: x e x e x Example: x x Example: x 0 e x 1 x (x 2 /2) x 3

7 Indeterminate Products If x a f(x) = 0 and x a g(x) = ±, then the it is an indeterminate form of type 0. [f(x)g(x)] x a This it may be 0,, or some compromise, a finite nonzero number. To solve this problem, write the product as a quotient; f(x)g(x) = f(x) 1/g(x) or f(x)g(x) = g(x) 1/f(x) The quotient will then have indeterminate form 0/0 or /. Example: x 0 x2 ln x +

8 Indeterminate Differences If x a f(x) = and x a g(x) =, then the it is an indeterminate form of type. [f(x) g(x)] x a This it is a contest between f and g. If f wins, the it is. if g wins, the it is. Or, there may be some finite number compromise. To determine this it, convert the difference into a quotient: common denominator, rationalization, factoring. The quotient will then have indeterminate form 0/0 or /. Indeterminate Powers There are several indeterminate forms that arise from the it x a [f(x)]g(x) 1. x a f(x) = 0 and x a g(x) = 0 type x a f(x) = and x a g(x) = 0 type 0 3. x a f(x) = 1 and x a g(x) = ± type 1 Each of these three cases can be solved by taking the natural logarithm: This will lead to an indeterminate product. Find the it of ln y = L. Then x a y = e L. let y = [f(x)] g(x), then ln y = g(x) ln f(x).

9 Example: 2x)1/x x 0 +(1 Example: x x1/x Example: ( 1 x 0 + x 1 ) e x 1

10 g replacements 2x Example: Consider (tan x)tan x π/4 Use a graph to estimate the value of the it. Use L Hôpital s Rule to find the exact value y π/8 π/4 3π/8 x Example: Let f(x) = e x 1 and g(x) = x 3 + 4x. Graph both f(x)/g(x) and f (x)/g (x) near x = 0. Find x 0 f(x) g(x). y g replacements x

11 Example: If an object with mass m is dropped from rest, one model for its speed v after t seconds, taking air resistance into account, is v = mg c (1 e ct/m ) where g is the acceleration due to gravity and c is a positive constant. (a) Find v. What is the meaning of this it? t (b) For fixed t, use l Hôpital s Rule to calculate v. What can you conclude about the speed m of a very heavy falling object? Example: Let S(x) = x 0 sin(πt 2 /2) dt Evaluate x 0 S(x) x 3.

12 Example: The figure below shows a sector of a circle with central angle θ. Let A(θ) be the area of the segment between the chord P R and the arc P R. Let B(θ) be the area of the triangle P QR. A(θ) Find θ 0 + B(θ). P g replacements A(θ) B(θ) θ O Q R

13 Example: Let A(t) be the area under the curve y = sin(x 2 ) from 0 to t. Let B(t) be the area of the triangle with vertices O, P, and (t, 0). A(t) Find t 0 + B(t). y P (t, sin(t 2 )) y P (t, sin(t 2 )) g replacements PSfrag replacements t x t x

14 Example: Suppose a and b are positive real numbers. Find x ( a 1/x + b 1/x ) x. 2

15 Example: A weight of mass m is attached to a spring suspended from a support. The weight is set in motion by moving the support up and down according to the formula h = A cos ωt, where A and ω are positive constants and t is time. If frictional forces are negligible, then the displacement s of the weight from its initial position at time t is given be where ω 0 = s = Aω2 ω 2 0 ω2 (cos ωt cos ω 0t), k/m for some constant k and with ω ω 0. Find ω ω 0 s. This shows the resulting oscillations increase in magnitude and is an example of the phenomenon of resonance.

16 Example: Suppose that n is a fixed positive integer. Find ( ) n x n + a n 1 x n 1 + a n 2 x n a 1 x + a 0 x x

17 Example: The surface area of an ellipsoid obtained by revolving the ellipse x2 a 2 + y2 = 1 (a > b) b2 around the x-axis is [ b A = 2πab a + a ( c c a)] sin 1 where c 2 = a 2 b 2. Find b a A.

18 Example: Find all positive integers n for which the given it exists. x 1 + e x 1 x (x 1) n