Lecture 7: Indeterminate forms; L’Hˆopitals rule; Relative rates of growth

1. Indeterminate Forms.

Example 1: Consider the limit x2 1 lim − . x→1 x 1 − 0 If we try to simply substitute x = 1 into the expression, we get . This is a so-called ³0´ indeterminate form. A limit of that form could be anything. After all, every derivative

′ f(x + h) f(x) f (x) = lim − h→0 h

is of that form (that is, a limit of this form could be any possible number). We have seen that in many cases algebra can be used to simplify the expressions to obtain a non-indeterminate form whose limit we can evaluate. In this example

x2 1 (x 1)(x + 1) lim − = lim − = lim x +1=2 x→1 x 1 x→1 x 1 x→1 − − We can do the division by x 1 in this example since we are only considering the limit as x approaches 1, but not what happens− at x = 1. Therefore, since x = 1, we can divide by x 1. 6 −

There are many possible indeterminate forms. They are

0 ∞ 0 0 , ∞ , , 0 , 1 , , 0 . ³0´ ³ ´ ³∞ − ∞´ ³ · ∞´ ³ ´ ³∞ ´ ³ ´ ∞ The actual value of these limits depends on how fast the respective numerators, denominators, basis, expo- nents and factors approach 1, 0, or . The following examples illustrate them: ∞ sin x 0 Example 2: lim (In Calculus I, we went through a rather complicated geometrical argument x→0 x ³0´ to show that this limit equals 1.)

ln x 0 Example 3: lim x→1 x 1 ³0´ − ex Example 4: lim ∞ x→∞ x ³ ´ ∞ 1 cos x 0 Example 5: lim − 2 x→1 x ³0´

2 −x ∞ Example 6: lim x e 0) Notice that you can rewrite this one in the form ∞ ). x→∞ ³∞ · ³ 1 Example 7: lim x ln x 0 ) x→0+ ³ · ∞

Example 8: lim (√x + 1 √x 1) We already know how to handle these limits. x→∞ − − ³∞ − ∞´ Hint: multiply top and bottom by the “conjugate”.

r ∞ Example 9: lim (1 + )n 1 Trick: When the variable is in the base and in the exponent, n→∞ n ³ ´ begin by taking the limit of the logarithm.

Example 10: lim xx 00 Same here. x→0 ³ ´

1 0 Example 11: lim x x ) x→∞ ³∞

1 ∞ Example 12: lim x x 0 ) Indeterminate? x→0+ ³

2. L’Hˆopital’s Rule.

0 ∞ L’Hˆopital’s rule is a tool to handle the case ³ 0 ) and ³ ∞ ). Theorem: (L’Hˆopital’s rule) If f(x) 0 and g(x) 0 then → → f(x) 0 f ′(x) lim = lim ′ x→a g(x) ³0´ x→a g (x)

If f(x) and g(x) then → ∞ → ∞ f(x) f ′(x) lim ∞ = lim ′ x→a g(x) ³ ´ x→a g (x) ∞

A proof of this theorem is outlined in the book. However, it is easy to see where this comes from using linear approximations, which we will learnt about last semester: If f is continuous and continuously differentiable at x = a, then one can approximate f(x) near x = a by its linearization:

′ f(x) f(a) + f (a)(x a) ≈ − This approximation gets better and better as x a. Same for g(x). Thus in the limit we can replace f and g by their approximations: → f(x) f(a) + f ′(a)(x a) lim = lim ′ − x→a g(x) x→a g(a) + g (a)(x a) − If f(a) = g(a) = 0 this expression simplifies to

f(x) f ′(a) lim = lim ′ x→a g(x) x→a g (a)

as long as g′(a) = 0. The fact that we can use this rule even if g′(a) = 0 requires a more careful proof. 6 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 approximations. In class we’ll apply L’Hˆopital’s rule to solve the examples above, where applicable.

2 3. Relative rates of growth. It is often important to determine how fast functions f(x) grow for very large values of x, and to compare the growth rate of various functions. Figure 1 Figure 2 9 900 x2 Ex 1: Any quadratic function grows faster than any linear 8 800 x2 function eventually. That is, even though for some 7 700 values of x the quadratic function may have smaller magnitude and grow slower than the linear function, 6 600

the quadratic growth will dominate the linear one if x 5 500 is large enough. (Compare x and x2, for example, as in 4 400 Figure 1.) 3 x 300 10x

2 200 Ex 2: We know that while the values of one linear function 1 100 may be larger than those of another, any two linear x functions eventually grow slower than any quadratic 0 0 0 1 2 3 0 10 20 30 function. (Figure 2 compares x, 10x and x2, as an example.)

Figure 3 Ex 3: You may have noticed that exponential functions like 300 x x 2 and e seem to grow more rapidly as x gets large ex than polynomials and rational functions. Figure 3 com- 250 2x pares ex and 2x with x2. You can see the exponentials 200 outgrowing x2 as x increases. In fact, as x , the x x → ∞ functions 2 and e grow faster than any power of x, 150 even x1,000,000. To get a feeling for how rapidly the values of y = ex 100 grow with increasing x, think of graphing the function x2 50 on a large blackboard, with the axes scaled in centime- 1 ters. At x = 1 cm, the graph is e 3 cm above the x- 0 0 1 2 3 4 5 6 7 8 axis. At x = 6 cm, the graph is e6 ≈ 403 cm 4m high (probably higher≈ than≈ the ceiling). At x = 10 cm, the graph is e10 22, 026 cm 220 m high, higher than most buildings. At x = 24 cm, the graph is more than halfway≈ to the moon,≈ and at x = 43 cm, the graph is high enough to reach past the sun’s closest stellar neighbor, the red dwarf star Proxima Centauri. Yet with x = 43 cm from the origin, the graph is still less than 2 feet to the right of the y-axis.

Here we want to compare, in particular, the growth rates of the new functions we have learned about (logarithms, exponentials), 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 x is large enough. For present purposes, we restrict our attention to functions whose values eventually become and remain positive as x . → ∞

3 Definition: Let f(x) and g(x) be positive for x sufficiently large. f(x) 1. f(x) grows faster than g(x) as x if lim = . → ∞ x→∞ g(x) ∞ f(x) 2. f(x) grows slower than g(x) as x if lim = 0. → ∞ x→∞ g(x) f(x) 3. f(x) and g(x) grow at the same rate as x if lim = L = 0, → ∞ x→∞ g(x) 6 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 limits involved we often use L’Hˆopital’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. xm grows slower than xn if m < n. ax grows slower than bx if a < b. Logarithms grow slower than polynomials which grow slower than (growing) exponentials.

Exercises:

1. (a) Show that x2 grows faster than x as x . → ∞ (b) Show that x and 10x grow at the same rate as x . → ∞

2. (a) Show that ex grows faster than x2 as x . → ∞ (b) Show that ex grows faster than 2x as x . → ∞

3. (a) Show that ln x grows slower than x as x . → ∞2 (b) Show that ln x grows at the same rate as ln(x ) as x . → ∞

2 4. Show that any quadratic function f(x) = a2x + a1x + a0 grows faster than any linear function g(x) = b1x + b2, where a2, b1 > 0, as x . → ∞ 4 5. Show that any two linear functions, f(x) = a0 + a1x and g(x) = b0 + b1x, a1, b1 > 0, grow at the same rate (namely linearly) as x . Similarly, one can show that any two polynomial functions of equal degree grow at the same rate.→ ∞

6. (a) Show that log (x) and log (x) grow at the same rate as x . a b → ∞ (b) Show that ax grows slower than bx as x , if a < b. For example, 2x grows slower than ex. → ∞ (c) Show that xa grows slower than xb if a < b. For example, √x grows slower than x which grows slower than x2.

7. Show that ln(x) grows at the same rate as ln(x3 3x + 1). − 5.⋆ Let n be a positive integer. (a) Show that ln(x) grows slower than xn as x . (a) Show that xn grows slower than ex as x →. ∞ → ∞ 8. Which of the following functions grow faster than x2? Which grow at the same rate as x2? Which grow slower? (a) x2 + 4x (h) 2x (b) √x (i) 10x2 (c) x2 + √x (j) (1.1)x (d) x3 x2 (k) (0.9)x −4 3 2 (e) √x x (l) log10 x (f) x ln x− (m) xx (g) x3e−x

9.⋆ Which of the following functions grow faster than ln x? Which grow at the same rate as ln x? Which grow slower?

(a) log3 x (g) 5 ln x (b) ln √x (h) ex 2 (c) x (i) log2(x ) (d) 1/x (j) log10(10x) (e) ln(2x) (k) ln(ln x) (f) √x (l) 10 ln x + x

10.⋆ Order the following functions from slowest growing to fastest growing, as x . Group functions that grow at the same rate together. → ∞ (a) ex (h) x2 (b) xx (i) √x (c) ln(x)x (j) (ln 2)x (d) ex/2 (k) ln(x2) 2 (e) 2x (l) ex (f) (0.9)x (m) x (g) 1/x2 (n) 1/x

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