Chapter 12

June 03 2017

Chapter 12 Indeterminate Forms and Improper Integrals 12.1 Introduction. The Mean Value Theorem - in the next few chapters we will need better methods for computing limits than any we have available now - our main purpose here are to understand the types of limit problems that lie ahead - and to acquire tools that will enable us to solve these problems with maximum efficiency - in section 2.5 we saw that the limit of a quotient is the quotient of limits - lim x>a f(x)/g(x) = L / M, provided that M != 0 - it is a fact of life that many of the most important limits are of the form in which both L = 0 and M = 0 - when this happens - the formula above is useless for calculating the value of the limit - the limit is said to have indeterminate form 0/0 at x = a - because the limit may very well exist - but nothing can be concluded about its vale without further investigation - for example - x/x, x2/x, x/x3, x sin 1/x / x - we see from these examples - by cancelling x's - such a quotient may have the limit 1, 0, or inf - or it may have no limit at all - indeterminate forms can sometimes be evaluated by using simple algebraic devices - by factoring and cancelling - in other cases - more complicated methods are required - for example - lim x>0 sin x / x - in section 9.2 a geometric argument was used to show that the value of this important limit is 1 - we point out the suggestive fact that this limit can also be evaluated by - noticing that it is the derivative of the function sin x at x = 0 - lim x>0 sin x / x = lim x>0 sin x - sin 0 / x - 0 = d/dx sin x ] x=0 = cos x ] x=0 = cos 0 = 1 - indeed - f'(a) = lim x>a f(x) - f(a) / x - a - is a version of the derivative definition equivalent to the usual - f'(x) = lim dx>0 f(x + dx) - f(x) / dx - every such derivative is an indeterminate form of the type 0/0 - these remarks suggest that there is a close but hidden connection between indeterminate forms and derivatives - this connection is understood through the Mean Value Theorem 12.2 The Indeterminate Form 0/0. L'Hospital's Rule - we begin to explore the connection between indeterminate forms and derivatives - by the simple theorem - if f(x) and g(x) are both 0 at x = a - and have derivatives there - then - lim x>a f(x) / g(x) = f'(a) / g'(a) = f'(x) / g'(x) ] x=a - provided that g'(a) != 0 - this requires the existence of derivatives at the single point a - if the derivatives exist in a interval about a and are continuous at a - then we can obtain the formula another way - by applying the Mean Value Theorem separately to the numerator and denominator - f(x) / g(x) = f(x) - f(a) / g(x) - g(a) = f'(c1) (x - a) / g'(c2) (x - a) = f'(c1) / g'(c2) > f'(a) / g'(a) - as x > a - the second alternative proof of the formula is important - because it often happens that the problems we consider have f'(a) = g'(a) = 0 - we can use the second proof to get around this difficulty - suppose c1 = c2 - then - f(x) / g(x) = f(x) - f(a) / g(x) - g(a) = f'(c) / g'(c) - where c is between x and a - in forming the limit as x > a - we can replace the quotient - f(x) / g(x) by f'(x) / g'(x) - L'Hospital's rule states that under certain easily satisfied conditions - this replacement is legitimate - lim x>a f(x) / g(x) = lim x>a f'(x) / g'(x) - provided the limit on the right exists - we should remember f(a) = g(a) = 0 is assumed here - as a practical matter - the functions we encounter in this book satisfy the conditions needed for L'Hospital's rule - we can apply the rule almost routinely - by continuing to differentiate the numerator separately as long as we still get the form 0/0 at x = a - as soon as one of these derivatives is different from zero at x = a - we must stop differentiating and hope to evaluate the last limit by some direct method 12.3 Other Indeterminate Forms - for certain applications it is important to know - L'Hospital's rule remains valid for indeterminate forms of inf/inf - can be used to show - lim x>inf xp / ex = 0 - for every constant p - this gives important insight into the nature of the exponential function - as x>inf - ex increases faster than any positive power of x - however large - therefore it grows faster than any polynomial - the powerful method of analysis based on L'Hospital's rule - extends easily to many similar limits - 0 inf, inf - inf, 00, inf 0, 1 inf 12.4 Improper Integrals - when we write down an ordinary definite integral as defined in chapter 6 - Sab f(x) dx - we assume the limits of integration are finite numbers - in chapter 14 it will be necessary to consider so-called improper integrals of the form - S a inf f(x) dx - we can define this integral as - S a inf f(x) dx = lim t>inf Sat f(x) dx - if this limit exists and has a finite value - then the improper integral is said to converge or to be convergent - it is remarkable that a region of infinite extent can have a finite area - another type of improper integral arises when the integrand f(x) is continuous on a bounded interval a <= x < b - but becomes infinite as x approaches b - in this case we can integrate from a to a variable upper limit t which is less than b - this integral is a function of t - we can now ask - whether this function approaches a limit as t>b - if so - we use this limit as the definition of the improper integral of f(x) from a to b - Sab f(x) dx = lim t>b Sat f(x) dx - as before - this integral is called convergent if the limit exists and is finite - and divergent otherwise