Math 19: Calculus Winter 2008 Instructor: Jennifer KlokeLecture Outline (Limits at Infinity)Monday, March 10Announcement• Office hours: March 11 (Tuesday) 7pm-10pm, and March 16 (Sunday) 7pm-10pm.They are tentatively scheduled in 380-T. In case they will be moved to a differentroom, a note will be posted on the door of 380-T.• In-class review session on Friday.• Homework will still be assigned this week. However, they will neither be collected orgraded. They are solely for your preparation for the final exam.RecapLast time, we talked about the correspondence between infinite limits and vertical asymp-totes. Remember vertical asymptotes only occur where the function is discontinuous. Afunction can have several vertical asymptotes.MotivationsExample 1. Consider the function f (x) =1x.We can observe the following from its graph:1. As x is getting larger and larger positively, the values of1xgets closer and closer to 0.Symbolically, we write limx→∞1x= 0.2. Similarly, as x is getting larger and larger negatively, the values of1xgets closer andcloser to 0. Symbolically, we write limx→−∞1x= 0.3. We say the line y = 0 (i.e. x-axis) is a horizontal asymptote, which indicates that theline is approached by the graph of the function when the magnitude of x is gettinglarge.Example 2. Consider the function f (x) = ex.1We can observe the following from its graph:1. As x is getting larger and larger negatively, the values of exgets closer and closer to0. So we say limx→−∞= 0.2. As x is getting larger and larger positively, the function exgrows unboundedly in thepositive direction. We say limx→∞= ∞.3. The limit of exwhen x → −∞ tells us that y = 0 is a horizontal asymptote.Example 3. Consider the function f (x) = x3.We can observe the following from its graph:1. limx→∞x3= ∞.2. limx→−∞x3= −∞.3. Since the function doesn’t approach any specific numb er when the magnitude of x islarge, the function doesn’t have any horizontal asymptote.Example 4. Consider the function f (x) = sin x.We can observe the following from its graph:1. As x gets larger and larger positively, the function is always oscillating between −1and 1. It neither approaches a certain number, nor does it grow unboundedly. In thiscase, we can only say limx→∞sin x does not exist.2. Similarly, in the other direction, we say limx→−∞sin x does not exist.3. Obviously, there’s no horizontal asymptote.Example 5. Consider the function f (x) = arctan x.We can observe the following from its graph:1. limx→∞arctan x =π2.2. limx→−∞= −π2.3. This is an example of a function which has two horizontal asymptotes. Since both y =π2and y =−π2are approached by the graph, both of them are horizontal asymptotes.2SummaryThe behavior of a function f(x) as x approaches ∞ has three different cases:1. It approaches a certain finite number L. We say limx→∞f(x) = L, and y = L is ahorizontal asymptote.2. It grows unboundedly in either the positive or the negative direction. We say limx→∞f(x) =∞ or − ∞.3. It doesn’t belong to either of the above cases. We can say no more than limx→∞f(x) doesnot exist.Similarly, we can talk about the behavior of a function as x approaches −∞. Theconclusion is completely parallel to above.ComputationsIn order to do algebraic computations without looking at graphs, we have the following 4principles:1. We can always make use of the known example limx→±∞1x= 0.2. Limit laws are still valid: taking limits commutes with addition, subtraction, multipli-cation, division, taking power and taking root.3. Usual techniques in limit computations are still applicable. For example, rationalizingnumerators.4. One more supplementary technique: divide both the numerator and denominator of afraction by the highest power of x appearing in the denominator. See examples below.Example 6. Compute limx→∞x2+ 3x2− x.Solution. We need to divide both the numerator and the denominator by the highest powerof x appearing in the denominator, which is x2.limx→∞x2+ 3x2− x= limx→∞(x2+3x2)(x2−xx2)= limx→∞(x2x2+3x2)(x2x2−xx2)= limx→∞1 +3x21 −1x=limx→∞(1 +3x2)limx→∞(1 −1x)=1 + 3( limx→∞1x)21 − ( limx→∞1x)=1 + 3 · 021 − 0= 13Example 7. Find all horizontal asymptotes of the function f (x) =x2+ 3x2− xSolution. In order to find horizontal asymptotes, we only need to compute the limits of thefunction as x → ∞ or x → −∞. If the results are finite numbers, the function has horizontalasymptotes.We have computed in the previous example that limx→∞x2+ 3x2− x= 1. Using the same method,you can find limx→−∞x2+ 3x2− x= 1 (do the computation on your own!). Therefore, y = 1 is theonly horizontal asymptotes of the function.Example 8. Compute limx→∞(√x2+ 1 − x).Solution. We need to rationalize the numerator first. The denominator in this problem isimplicitly given by 1. After that we need to divide both the numerator and the denominatorby the highest power of x in the denominator.limx→∞(√x2+ 1 − x) = limx→∞(√x2+ 1 − x)(√x2+ 1 + x)(√x2+ 1 + x)= limx→∞x2+ 1 − x2√x2+ 1 − x= limx→∞1√x2+ 1 + x= limx→∞(1x)(√x2+1+xx)= limx→∞1x√x2+1x+xx= limx→∞1xqx2+1x2+ 1= limx→∞1xq1 +1x2+ 1=limx→∞1xr1 + limx→∞1x2+ 1=0√1 + 0 + 1= 0UpcomingOn Wednesday, we are gonna do our last new topic which is l’Hospital’s rule. There is aninteresting story about this rule which says this rule is actually discovered by Bernoulli. Whyis it under the name of l’Hospital? Read the story on page 305 of