**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