Lecture for Week 3 (Secs. 2.5 and 2.6)Infinity and Continuity(inclu ding vertical asymptote s from Sec. 2.2)1Infinity is not a nu mber. It is a figureof speech.Stewart pp. 86–88 and 109 shows graphicalexamples of functions with vertical asymptotes,or, equivalently, infinite limits. The asymptote isa vertical line whi ch is approached by t he graph .Typically, it appears at a spot where the de nom-inator of the function vanishes (equals 0) but thenumerator is not zero.2limx→af(x) = +∞means t hat the values of f (x) can be forced tobe arbitrarily large (and positive) by consideringonly x s ufficiently close to a.This definition doesn’t allow x to be actuallyequal to a. Usually f(a) is not even defined.3Note that a function likesin1xxhas arbitrarily lar ge values near a = 0, but it doesnot approach +∞, because it also has small valuesarbitraril y close to a . (For instance, it’s 0 whenx =1Nπ.)The limit is − ∞ if the fun ction becomes ar-bitrarily large and negative around the asymp-tote.4Often a functi on will approach +∞ from oneside an d −∞ from the other:Exercise 2.2.18 (p. 90), extendedDiscus s the behav ior of f (x) =6x − 5aroundx = 5.5As x → 5 from the right, the denominatorbecomes very s mall but remains positive:limx→5+6x − 5= +∞.As x → 5 from the left, the denominator becomesvery small and negative:limx→5−6x − 5= −∞.The line x = 5 is a vertical asymptote.6In Sec. 2.6 we have graphs with horizontalasymptotes, represe nting functions with definitelimits at infinity. The asympt ote is a horizon -tal line that t he graph approache s at either theextreme right of the graph , or the ext re me left,or both. Notice (last pi cture on p. 123) th at thegraph d oes not need to stay on one side of theasymptote — it can wiggle around it.7For example,limx→−∞f(x) = Lmeans t hat the values f (x) can be forced to bearbitrary close to L by considering only values ofx t hat are sufficie ntly large an d negative. (Moreprecise ly, I should say “negative and sufficientlylarge in absol ute valu e.”)8Exercise 2.6.33 (p. 133)limx→∞x + 4x2− 2x + 5Exercise 2.6.17limx→∞(√1 + x −√x)9limx→∞x + 4x2− 2x + 5is a ration al fu nction (ratio of two polynomials).The basis trick for finding limits of such thin gsat in finity is:Divide numerator and denominator bythe high e st power appearing in the denom-inator.10x + 4x2− 2x + 5=1x+4x21 −2x+5x2.The point of t his maneuver is that now the de-nominator approaches 1 as x → ∞, so all weneed to do is to take the limit of t he numerator,which is 0 in this case.limx→∞x + 4x2− 2x + 5= 0.11It’s easy to see what will happen in al l prob-lems of thi s type.1. If t he denominator has highe r degree thanthe numerator, the limit is 0.2. If t he nume rator and denominator have thesame degree, the limit is some nonz ero nu m-ber ( t he coefficient of the leading term inthe numerator).3. If t he nume rator has the higher degree, the12two limits at infini ty are infinite (possibl y ofopposite signs). Example:x3+ 3x2− 1=x +3x21 −1x2,limx→+∞x3+ 3x2− 1= +∞, limx→−∞x3+ 3x2− 1= −∞.With rational functions we can’t constructan example for which limx→+∞and l imx→−∞arefinite and differ ent. But the inverse t rig functiontan−1x has that property (see gr aph p. 279).13In a problem likelimx→∞(√1 + x −√x)it helps to “rational ize the numerator” by mu lti-plying numerator and denominator by the “con-jugate” expression, in this case√1 + x+√x. Youget(1 + x) − x√1 + x +√x.The numerator now goes to 1 while the den omi-nator goes to infi nity, so the limit is 0.14In both of our exercise examples , the hori-zontal axis y = 0 was a horizontal asymptote .Here is an example with a different result:Exercise 33, extendedFind any asymptotes of y = f(x) =xx+4, andstate the corresponding l imits involving infinity.15There is a vertical asymptote at x = −4.limx→−4+f(x) = −∞, limx→−4−f(x) = +∞.limx→±∞f(x) = lim11 +4x= 1.There is a horizontal asymptote at y = 1.16Now, what is continuity?The intuitive idea is that a function is c on-tinuous if its graph can be drawn in one stroke,never lifting the pencil from the paper...............................................................................................................................................................................................................................................................................................................................................................................xy17The practical meaning of continuity for do-ing calculations is related to something we talkedabout last we ek. Remember that the crucialpractical question about limits is, when do weknow thatlimx→af(x) = f(a) ?More generally, in evaluat ing li mits we oftenwanted to “push the limit through” a function:18limu→bf(g(u) ) = flimu→bg(u)?If limu→bg(u) exis t s and equals a, then t hat ma-neuver is correct, provided thatlimx→af(x) = f(a). (∗)If (∗) is true , we say that “f is continuous at a”— a ve ry convenient property for a func t i on tohave!19If a function fail s t o be cont inuous at a, thenone of three things has happened:1. f(a) is not de fined.2. limx→af(x) does not exi st.3. Those two numbers exist but are n ot equal.(It is possible for bot h 1 and 2 to happen at once.)20The limits that define derivatives, of thetypelimx→ag(x) − g(a)x − a,are th e classic ex ampl e of discontinuous function sof type 1. They are the main reason for studyinglimits at the beginning of cal culus.We have seen various examples of type 2, in-cluding (a) vertical asymptotes; (b) points whereleft an d right limits exist but are not equ al .21It is easy to draw graphs of functions oftype 3: take a continuous funct ion and move onepoint on it vertically to a strange plac e.............................................................................................................................................. .............................................................................................................................................◦•It is harder to find such functions in “reallife”, but here’s an attempt. Suppose you shoota gun at a target containing a hole exactly the sizeof a bullet. If your aim is perfect, the bullet landson the far side. Otherwise, it ricochets off the
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