Section. Limits involving infinit (/3/08) Overview: In later chapters we will need notation and terminolog to describe the behavior of functions in cases where the variable or the value of the function becomes large. We sa that or tends to if it becomes an arbitraril large positive number and that or tends to if it becomes an arbitraril large negative number. These concepts are the basis of the definitions of several tpes of its that we discuss in this section. Topics: Infinite its as ± Finite its as ± One-sided and two-sided infinite its Infinite its of transcendental functions Infinite its as ± Imagine a point that moves on the curve = 3 in Figure. As the -coordinate of the point increases through all positive values, the point moves to the right and rises higher and higher, so that it is eventuall above an horizontal line, no matter how high it is. We sa that 3 tends to as tends to and write 3 =. = 3 8 FIGURE Similarl, as the -coordinate of the point decreases through all negative values, the point moves to the left and drops lower and lower so that it is eventuall beneath an horizontal line, regardless how low it is. We sa that 3 tends to as tends to, and we write 3 =. The function = 3 illustrates the first and fourth parts of the following definition. Definition (Infinite its as tends to ± ) (a) f() = if f() is an arbitraril large positive number for all sufficientl large positive. (b) f() = if f() is an arbitraril large negative number for all sufficientl large positive. (c) f() = if f() is an arbitraril large positive number for all sufficientl large negative. (d) f() = if f() is an arbitraril large negative number for all sufficientl large negative. When we sa that a negative number or is large, we mean that its absolute value is large. 36
Section., Limits involving infinit p. 37 (/3/08) Parts (a) and (b) of this definition appl onl if f is defined on an interval (a, ) for some number a, and parts (c) and (d) appl onl if f is defined on (, b) for some b. We can often determine the tpes of its described in Definition from the graphs of the functions, as in the net eample. Eample Solution What are and? The graph in Figure shows that = and =. = FIGURE The net eample illustrates a basic principle: an polnomial has the same its as and as as its term involving the highest power of. Eample Solution Find ( 3 ) and ( 3 ). We can epect that the its as ± of 3 will be those of its highest degree term, so that ( 3 ) = ( ) = ( 3 ) = ( ) =. () To verif these conclusions, we factor out the highest power of 3 b writing for 0, ( 3 = ). This quantit tends to as and as because / tends to and tends to. Properties () of the function = 3 can also be seen from its graph in Figure 3. 00 = 3 FIGURE 3
p. 38 (/3/08) Section., Limits involving infinit Finite its as ± The function = /( + ) of Figure has a different sort of behavior for large positive and large negative. Because /( + ) is ver small for large positive or negative, the value /( + ) approaches and the graph approaches the horizontal line = as and as. We sa that the it of /( + ) as tends to or to is, and we write ( ) ( + = and ) + =. = = + FIGURE Here is a general definition of this tpe of it: Definition (Finite its as tends to ± ) positive. (a) (b) negative. If f() = L with a number L if f() is arbitraril close to L for all sufficientl large f() = L with a number L if f() is arbitraril close to L for all sufficientl large f() = L or f() = L, then the line = L is a horizontal asmptote of the graph. The line =, for eample, is a horizontal asmptote of the graph in Figure. Eample 3 Calculate the values of f() = 3 + at = ±00, ±000, and ± 0,000 and use + 3 the results to predict the its of + as and as. + 3 Solution The values in the table below suggest that + = and + 3 + + =. 0,000 000 00 00 000 0,000 3 +.99999998.99999800.99979997.99979997.99999800.99999998 + The its as ± of a quotient of polnomials or other linear combinations of powers are the its of the quotient of the highest-degree terms. If the numerator and denominator have the same degree, as in Eample 3, this can be verified b dividing the numerator and denominator b n, where n is the degree of the numerator and denominator.
Section., Limits involving infinit p. 39 (/3/08) Eample Solution Find 3 + and + 3 + +. Since the term of highest degree in the numerator is and the term of highest degree in the denominator is, we can anticipate that ± 3 + + = ± = ( ) =. ± To verif this, we divide the numerator and denominator of the given function b to obtain for 0, 3 + + = 3 + + Then, since 3/, / 3, and / tend to 0 as ±, ± 3 + + = =. The line = is a horizontal asmptote of the graph (Figure 5).. = 3 + + FIGURE 5 = If the numerator and denominator of a quotient of polnomials are of different degrees, we find the its of the quotient as ± b dividing the numerator and denominator b n, where n is the lower of the two degrees.
p. 0 (/3/08) Section., Limits involving infinit Eample 5 3 + Find + and 3 + +. Solution Since the terms of highest degree in the numerator and denominator are 3 and, respectivel, we anticipate that 3 + + = 3 = 3 = 0. The numerator is of degree and the denominator is of degree, so to verif this conclusion we divide both b =. We obtain 3 + + = 3 + = 0. + The last epression tends to 0 as tends to because the numerator 3 + tends to 3 and the denominator + tends to. B the same reasoning, 3 + + = 3 + = 0. + Eample 6 Solution 3 What are + and 3 +? We anticipate that = 3 3 has the same its as ± as = =, so that + it tends to as and tends to as. To verif this, we divide the numerator and denominator b to obtain 3 + = for 0. + Since and tend to 0 as ±, the last equation shows that 3 + = + = 3 + = =. + One-sided and two-sided infinite its Figure 6 shows the graph of = /. Because its values become arbitraril large positive numbers as approaches 0 from the right and become arbitraril large negative numbers as approaches 0 from the left, we sa that = / tends to as tends to 0 from the right and tends to as approaches 0 from the left. We write 0 + = and 0 =. In contrast, the values of = / in Figure 7 become arbitraril large positive numbers as approaches 0 from either side, so we sa that = / tends to as tends to 0. We write 0 =.
Section., Limits involving infinit p. (/3/08) = = FIGURE 6 FIGURE 7 These functions illustrate the following definition: Definition 3 (Infinite one-sided and two-sided its) to a. to a. to a. to a. (a) f() = if f() is an arbitraril large positive number for all > a sufficientl close a + (b) f() = if f() is an arbitraril large positive number for all < a sufficientl close a (c) f() = if f() is an arbitraril large negative number for all > a sufficientl close a + (d) f() = if f() is an arbitraril large negative number for all < a sufficientl close a (e) f() = if f() = and f() =. a + a a (f) f() = if f() = and f() =. a + a The line = a is a vertical asmptote of = f() if f() or f() as a + or as a. The -ais ( = 0), for eample, is the vertical asmptote of = / in Figure 6 and of = / in Figure 7. a
p. (/3/08) Section., Limits involving infinit Eample 7 A compan produces a chemical at a cost of $0 per liter plus a dail overhead (fied cost) of $000. (a) What is the compan s total cost C = C() to produce liters in a da? (b) What is the average total cost A = A() per liter if liters are produced in a da? (c) Find A() and A() and eplain the results. 0 + Solution (a) It costs [0 dollars per liter][ liters] = 0 dollars to produce liters, plus 000 dollars overhead, so C() = 000 + 0 (dollars). The graph of this function is the line in Figure 8. 5000 000 3000 000 000 C (dollars) C = C() 80 60 0 0 A (dollars per liter) A = A() A = 0 50 00 50 50 00 50 (liters) C() = 000 + A() = 000 + 0 (liters) FIGURE 8 FIGURE 9 (b) The average cost per liter is A() = C() dollars liters = 000 + 0 = 000 dollars + 0 liter. () (c) As can be seen from () and A() = A() = 0 + [ 000 + 0 ] = 0 dollars (3) [ 000 + 0 ] =. () Consequentl, the graph of the average cost in Figure 9 has the line A = 0 as a horizontal asmptote and the A-ais as a vertical asmptote. The average cost per liter A() is close to 0 for large, as indicated b (3), because if is large, the overhead is spread out over a large number of liters of the chemical and the average cost is close to the cost of one liter with no overhead. The average cost is ver large for ver small, as indicated b (), because for small the overhead has to be covered b a small volume of the chemical.
Section., Limits involving infinit p. 3 (/3/08) Eample 8 Find 0 ( + 3 ). Solution Since 3 tends to 0 as 0, 0 ( + 3 ) = = 0 0 =. This propert can be seen from the graph of = + 3 in Figure 0. = + 3 FIGURE 0 0 A quotient = p()/q() of polnomials has a vertical asmptote at an a such that q(a) = 0 and p(a) 0 since the denominator tends to 0 and the numerator tends to the nonzero number p(a) as a. The behavior of the graph on both sides of the asmptote can be found b determining whether the function is positive or negative for slightl greater than a and for slightl less than a. Eample 9 3 3 3 Find (a), (b), and (c) +. Solution (a) The graph = 3 has a vertical asmptote at =, where the denominator is 0 and the numerator is not zero. Because 3 and are positive for >, = 3 3 is positive for >. Therefore, + =. (b) On the other hand, 3 is positive and is negative for 0 < <. Consequentl, 3 = is negative for 0 < <. Consequentl, are shown b the graph of the function in Figure. (c) The two-sided it are different. + 3 3 =. These its = is not defined because the one sided its If p(a) = 0 and q(a) = 0, then a power of a must be factored from the numerator and denominator and cancelled so that the new numerator and/or the new denominator are not zero at a before ou can determine whether the graph has a vertical asmptote at = a.
p. (/3/08) Section., Limits involving infinit = 3 5 0 5 FIGURE 3 5 Infinite its of transcendental functions The eponential function = e tends to as tend to 0 as (Figure ). The logarithm = ln tends to as 0 + and tend to as (Figure 3). = e = ln 6 FIGURE FIGURE 3 The net eample deals with a function constructed from = e. Eample 0 Figure shows the curve = e /. Eplain wh (a) ± e/ =, (b) 0 + e/ =, and (c) 0 e/ = 0. = e / FIGURE 6 Solution (a) ± e/ = because z = / 0 as ± and z 0 e z = e 0 =. (b) 0 + e/ = because z = / as 0 + and z ez =. (c) e/ = 0 because z = / as 0 + and 0 z ez = 0.
Section., Limits involving infinit p. 5 (/3/08) The trigonometric functions do not have its as ± because the are periodic. The inverse sine and cosine functions do not have its as ± because the are defined onl for. The inverse tangent function, in contrast, tends to π as and tends to π as (Figure 5). = π = tan FIGURE 5 = π The tangent, cotangent, secant, and cosecant functions have infinite one-sided its at the points where their denominators are zero. Other functions constructed from trigonometric functions can also have infinite its, as in the net eample. Eample Solution cos cos Find (a) and (b) 0 + 0. (a) For small positive, cos is close to cos(0) = and / is a large positive number, cos so =. 0 + (b) For small negative, cos is close to cos(0) = and / is a large negative number, cos so =. These properties can be seen from the graph in Figure 6. 0 = cos 0.5 π FIGURE 6
p. 6 (/3/08) Section., Limits involving infinit Eample Solution Find (a) tan (e ) and (b) tan (e ). (a) tan (e ) π as because e as and tan π as. (b) tan (e ) 0 as because e 0 as and tan 0 as 0 (Figure 7) = tan (e ) = π FIGURE 7 Interactive Eamples. Interactive solutions are on the web page http//www.math.ucsd.edu/ ashenk/.. Find (a) f() and (b) f() for f() = 3. C (c) Generate the graph of f in the window, 3 3 as a partial check of our answers. 3. What are the its of g() = + (a) as and (b) as? C (c) Generate the graph of g in the window 6 9, 5 5 as a partial check of our answers. 3. Find (a) +. Find 3 ( + ). and (b). 5. What are the its of = ln (a) as 0+, (b) as (d) as? (See Figure 8.) (c) as + and = ln 3 FIGURE 8 In the published tet the interactive solutions of these eamples will be on an accompaning CD disk which can be run b an computer browser without using an internet connection.
Section., Limits involving infinit p. 7 (/3/08) Eercises. A Answer provided. CONCEPTS: O Outline of solution provided. C Graphing calculator or computer required.. What is [f() 00] if f() =? f() = 0?. What is [f() 00] if 3. What is [00 f()] if f() =?. What is f() 5. What is 0 + f() if f() = 0 and f() > 0 for > 000? if f() = 0 and f() < 0 for 0 < < 0.000? 0 + 6. Suppose that = is an asmptote of = f(). Wh is = also an asmpote of = [f()]? 7. (a) Eplain the shape of the graph = sin in Figure 9. (b) Use Definitions a and b to eplain wh = sin does not tend to or as. 0 = 0 0 = sin FIGURE 9 0 = BASICS: 8. O Figure 0 shows the graph of the polnomial p() = 3 + 5 + Use the formula to find (a) p() and (b) p(). = p() 0 FIGURE 0 0
p. 8 (/3/08) Section., Limits involving infinit 9. O Figure shows the graph of and (c) f().. Use the formula to find (a) f(), (b) ( ) 6 = f() f(), FIGURE 0. O 0 5 What is +? 6. O Find () and 5 (), where () = +.. O What is + 6? 3. O Find (a) + and (b) 3 ( ) ( ) 3. In Eercises through use the formulas to find (a) the graphs in the given windows as a partial check of our answers.. O f() = 3 ( 5, 0 5) 5. O f() = 3 3 ( 3 3, ) 6. A f() = + 3 3 (, ) 7. f() = 3 + 3 3 ( 5, 3 3) 8. O f() = + 9. A f() = 0. f() = + (,.5.5) ( 5 0, 5 0) (, 6 8) f() and (b) f(). C (c) Generate. f() = 3 ( 5, 0 0) In Eercises through 5 use the formulas to find (a) f(), (b) f(), (c) f(), 0 + (d) f(), and (e) f(). C (f) Generate the graphs in the given windows as a partial check of 0 0 our answers.. O f() = + / ( 3 3, 8) 3. f() = + ( 3 3, 8 8). A f() = ( 3 3, 5 5)
Section., Limits involving infinit p. 9 (/3/08) 5. f() = 3 ( 3 3, 0 0) 6. O Find ( ). 7. A + 5 Find 0 + 0 and + 5 0 0. 8. Find 3 5 ( 3). 9. A What is cos(e )? 30. Find 0 + tan (/). EXPLORATION: In Eercises 3 and 3 use the formulas to find (a) f(), (b) f(), (c) f(), 0 + (d) f(), and (e) f(). C (f) Generate the graphs in the given windows as a partial check of 0 0 our answers. 3. O f() = ( + /)(3 5/ ) ( 6, 6 8) ( ) 3. f() = + ( ) + (, 5 5) 33. A What is 0 F() + F() + 5 3. Find ( + ). if F() =? 0 35. A What are the ampototes of = e + e? 36. Eplain wh sec = (a) b using properties of = cos and (b) b using (π/) Figure with the definition of = sec from trigonmetr. FIGURE
p. 50 (/3/08) Section., Limits involving infinit 37. The curve = tan in Figure 3 has two nonhorizontal, nonvertical asmptotes. What are the? 8 FIGURE 3 FIGURE 38. Find the its of = tan (/) in Figure as ± and as 0 ±. ( ) ( ) 39. Which of the curves (a) = cos + (b) = sin + is in Figure 5 and which is in Figure 6? Eplain. FIGURE 5 FIGURE 6 0. What are the asmptotes of = ln ln?. What are the asmptotes of = (ln ) 0ln + 5?. 3 Find + + 3. What is the number C if C cos is a number L? (End of Section.)