Lecture 7: Indeterminate forms; L Hôpitals rule; Relative rates of growth. If we try to simply substitute x = 1 into the expression, we get

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1 Lecture 7: Indeterminate forms; L Hôpitals rule; Relative rates of growth 1. Indeterminate Forms. Eample 1: Consider the it If we try to simply substitute = 1 into the epression, we get. This is a so-called indeterminate form. A it of that form could be anything. After all, every derivative f ) = h f + h) f) h is of that form that is, a it of this form could be any possible number). We have seen that in many cases algebra can be used to simplify the epressions to obtain a non-indeterminate form whose it we can evaluate. In this eample = 1) + 1) = + 1 = We can do the division by 1 in this eample since we are only considering the it as approaches 1, but not what happens at = 1. Therefore, since 1, we can divide by 1. There are many possible indeterminate forms. They are,, ), ), 1,,. The actual value of these its depends on how fast the respective numerators, denominators, basis, eponents and factors approach 1,, or. The following eamples illustrate them: Eample : sin Eample 3: 1 ln 1 In Calculus I, we went through a rather complicated geometrical argument to show that this it equals 1.) e Eample 4: 1 cos ) Eample 5: 1 Eample 6: e Notice that you can rewrite this one in the form ). 1

2 Eample 7: ln + Eample 8: + 1 1) ) We already know how to handle these its. Hint: multiply top and bottom by the conjugate. Eample 9: 1 + r n n )n 1 Trick: When the variable is in the base and in the eponent, begin by taking the it of the logarithm. Eample 1: Same here. Eample 11: 1 Eample 1: + 1 ) ) Indeterminate?. L Hôpital s Rule. L Hôpital s rule is a tool to handle the case ) and ). Theorem: L Hôpital s rule) If f) and g) then If f) and g) then f) a g) = a f ) g ) f) a g) = a f ) g ) A proof of this theorem is outlined in the book. However, it is easy to see where this comes from using linear approimations, which we will learnt about last semester: If f is continuous and continuously differentiable at = a, then one can approimate f) near = a by its linearization: f) fa) + f a) a) This approimation gets better and better as a. Same for g). Thus in the it we can replace f and g by their approimations: f) a g) = fa) + f a) a) a ga) + g a) a) If fa) = ga) = this epression simplifies to f) a g) = f a) a g a) as long as g a). The fact that we can use this rule even if g a) = requires a more careful proof. While this is only an outline of a proof for a special case it gives good intuition. It shows that L Hopitals rule is just an application of linear approimations. In class we ll apply L Hôpital s rule to solve the eamples above, where applicable.

3 3. Relative rates of growth. It is often important to determine how fast functions f) grow for very large values of, and to compare the growth rate of various functions. E 1: Any quadratic function grows faster than any linear function eventually. That is, even though for some values of the quadratic function may have smaller magnitude and grow slower than the linear function, the quadratic growth will dominate the linear one if is large enough. Compare and, for eample, as in Figure 1.) Figure Figure 1 E : We know that while the values of one linear function may be larger than those of another, any two linear functions eventually grow slower than any quadratic function. Figure compares, 1 and, as an eample.) E 3: You may have noticed that eponential functions like and e seem to grow more rapidly as gets large than polynomials and rational functions. Figure 3 compares e and with. You can see the eponentials outgrowing as increases. In fact, as, the functions and e grow faster than any power of, even 1,,. To get a feeling for how rapidly the values of y = e grow with increasing, think of graphing the function on a large blackboard, with the aes scaled in centimeters. At = 1 cm, the graph is e 1 3 cm above the - ais. At = 6 cm, the graph is e 6 43 cm 4m high probably higher than the ceiling). At = 1 cm, the graph is e 1,6 cm m high, higher than most buildings. At = 4 cm, the graph is more than halfway to the moon, and at = 43 cm, the graph is high enough to reach past the sun s closest stellar neighbor, the red dwarf star Proima Centauri. Yet with = 43 cm from the origin, the graph is still less than feet to the right of the y-ais. Figure 3 e Here we want to compare, in particular, the growth rates of the new functions we have learned about logarithms, eponentials), as well some of those we already know about polynomials, square roots, other powers). The following definition precisely states what it means for one function to grow faster than, grow slower than, or grow at the same rate as another one, eventually, that is, if is large enough. For present purposes, we restrict our attention to functions whose values eventually become and remain positive as. 3

4 Definition: Let f) and g) be positive for sufficiently large. f) 1. f) grows faster than g) as if g) =. f). f) grows slower than g) as if g) =. f) 3. f) and g) grow at the same rate as if g) = L, where L is some finite number. This definition implies that if f grows faster than g, then f will eventually be much larger than g. Similarly, if f grows slower than g, then f will eventually be much smaller than g. In order to compute the its involved we often use L Hôpital s rule. The notion of relative rates of growth will be very useful to us later on in this semester, when we talk about integrals over infinite domains and when we talk about series. In the problems below you will establish, among others, that: Any two polynomial functions of equal degree grow at the same rate. m grows slower than n if m < n. a grows slower than b if a < b. Logarithms grow slower than polynomials which grow slower than growing) eponentials. Eercises: 1. a) Show that grows faster than as. b) Show that and 1 grow at the same rate as.. a) Show that e grows faster than as. b) Show that e grows faster than as. 3. a) Show that ln grows slower than as. b) Show that ln grows at the same rate as ln ) as. 4. Show that any quadratic function f) = a + a 1 + a grows faster than any linear function g) = b 1 + b, where a,b 1 >, as. 4

5 5. Show that any two linear functions, f) = a + a 1 and g) = b + b 1, a 1,b 1 >, grow at the same rate namely linearly) as. Similarly, one can show that any two polynomial functions of equal degree grow at the same rate. 6. a) Show that log a ) and log b ) grow at the same rate as. b) Show that a grows slower than b as, if a < b. For eample, grows slower than e. c) Show that a grows slower than b if a < b. For eample, grows slower than which grows slower than. 7. Show that ln) grows at the same rate as ln ). 5. Let n be a positive integer. a) Show that ln) grows slower than n as. a) Show that n grows slower than e as. 8. Which of the following functions grow faster than? Which grow at the same rate as? Which grow slower? a) + 4 h) b) i) 1 c) + j) 1.1) d) 3 k).9) e) 4 3 l) log 1 f) ln m) g) 3 e 9. Which of the following functions grow faster than ln? Which grow at the same rate as ln? Which grow slower? a) log 3 g) 5ln b) ln h) e c) i) log ) d) 1/ j) log 1 1) e) ln) k) lnln ) f) l) 1ln + 1. Order the following functions from slowest growing to fastest growing, as. Group functions that grow at the same rate together. a) e h) b) i) c) ln) j) ln ) d) e / k) ln ) e) l) e f).9) m) g) 1/ n) 1/ 5