Roberto s Notes on Dierential Calculus Chapter : Limits and continuity Section Basic properties o its What you need to know already: The basic concepts, notation and terminology related to its. What you can learn here: Some basic properties that will allow the computation o some simple its now and more comple ones later. c and computing the it there are not the same Evaluating a unction at thing conceptually, but they turn out to coincide in most situations, especially when we are dealing with what I call boring its. This is a consequence o a act that I will irst state here in a very inormal way, then present, without proo, in a list o basic properties. Knot on your inger: The anti-murphy s Law o its Unless a special eature o a unction complicates matters at a given value, a it may be computed by analyzing each piece o the unction separately. In other words, i there is no indication that something may go wrong with a unction at a value, nothing will go wrong. I call this the anti-murphy s law to relate it to the more common Murphy s law: I anything can go wrong, it will. For its, as long as there are no complicating actors, the it can be computed by breaking the unction down into its components and computing them or thinking through them. But be careul! Warning bells Sometimes the eature that makes a it worth computing, and hence problematic, may not be obvious or clearly visible, especially i you are not amiliar with the unction in question. Students oten have certain algebra misconceptions that lead them to errors. In such cases the real Murphy s Law may actually come into eect, with disastrous consequences on the accuracy o your mathematical work. Also, I am probably the only one that calls this property the anti-murphy s law, so do not epect to ind this epression in the literature, and do not epect others to know what it means! But or my students it seems to work as a way to convey the concept. So, use it i it helps you, but do not eel obliged to memorize the idea. Dierential Calculus Chapter : Limits and continuity Section : Basic properties o its Page
In a more ormal way, here are some o the basic, practical implementations o this concept. You will see some more among the Learning questions. Technical acts: Limit laws and I a g b, with both a and b c c real numbers, then: g c g a b c c g c c c g ab c a as long as b 0 c g g b c g g b c a as long as c c a and b are not both 0. Remember that these acts can be a double-edged sword: they can make many calculations easy, but they can also lull you into a alse sense o security and the wrong conviction that any simpliying operation can be done on its. We ll spend some time in later sections analyzing the subtle situations were things can go wrong. For now, watch out! But let us start by looking at a situation when the laws simpliy our lie. Eample: cos This may seem like a complicated it, but when we look at the unction, we realize that there is nothing wrong with it at or around. So, we break it down into its parts: 6 64 cos cos cos There is another important aspect o its that is linked to a situation where a it does not eist, but we can say something more and useul about why. I Deinition can become arbitrarily large as approach c, we say that the unction approaches ininity, or that its it at c is ininite and write: c i the values o ever larger and positive as c c. i the values o become ever larger and negative as The same applies to one-sided its. become c. There are two important issues to keep in mind about ininite its. Dierential Calculus Chapter : Limits and continuity Section : Basic properties o its Page
Warning bells From the technical point o view, i a it is ininite, it does not eist. The notation is meant to provide c useul inormation about the behaviour o the unction, NOT to assert the eistence o a it. Warning bells The symbols and - represent concepts and not numbers. They represent an interesting iting behaviour o a unction. Thereore, usual algebraic rules do NOT apply to them and we cannot use the anti-murphy s law when it involves ininite its. Eample: This time the denominator again approaches 0, but it is positive to the right o and negative to the let o. So we can use the same logical process on each side separately: ; However, notice that this time the two its provide two dierent types o ininity. Thereore we cannot say that the two-sided it is ininite: which ininity would it be? Instead we conclude that such it does not eist: DNE Again, the calculator s graph conirms this conclusion. The last two eamples are instances o another basic and very useul property o its, one to which I have also given a special, but not commonly used name. Eample: This is a raction whose denominator approaches 0, so we cannot use the anti- Murphy s law, as we cannot divide by 0. However we notice that by letting approach, this denominator can be made arbitrarily small and positive so that the whole unction can be made arbitrarily large and positive, as the graph shows. We can indicate this by writing: I Technical act: The it law o balloons and a 0 c c g c g i i g 0, then c a 0 g as c. a 0 g as c. Dierential Calculus Chapter : Limits and continuity Section : Basic properties o its Page
This situation is usually reerred to as a #/0 orm and applies to one-sided its as well. So, why do you call it the law o balloons? Because it evokes the image o balloons in the sense that as c, the denominator becomes very small, hence very light, thus pushing the whole raction up to ininity! It is an image that appeals to me and to some students, but not to everyone. Use it only i it helps you. Eample: cosh cos This looks like a comple it, but again, we can think our way through it. The hyperbolic cosine unction always generates positive values, so, whatever cosh is, it is positive. On the other hand cos 0, but it approaches 0 rom dierent sides. Thereore, we need to look at the one-sided its separately: cosh # cos 0 cosh # cos 0 I assume and hope that the notation I just used to indicate whether we are dealing with positive or negative numbers is clear. The graph conirms this conclusion. There is another useul and practical property that has earned a special name rom me. Technical act: The it law o gravity and I h g (meaning c c that both options are acceptable), then: 0 c g This is usually reerred to as a #/ orm and applies to one-sided its as well. I use the image o gravity because here the denominator is becoming large think o it as heavy thus pushing the unction into the ground, at level 0. As always, use the image at your discretion. Eample: cos 0 ln We know rom the properties o these unctions, that the numerator approaches and the denominator approaches -. Thereore we can easily conclude that this it is 0. And once again, the graph supports the conclusion, even though the graph itsel is not totally convincing. But this time we have logic on our side. Dierential Calculus Chapter : Limits and continuity Section : Basic properties o its Page 4
Finally, there is another important property that has a strange name, but this time it is a name given by someone else and recognized by the whole mathematical community. Technical act: The Squeeze Theorem Assume that c a, b that: or every a, b then:, meaning that a c b, and g ecept possibly or c. I c c L and L g L c Although this theorem and its name are traditional items in a calculus course, it is mainly a technical theorem, with ew practical applications o note. This is because usually proving the inequality among the three required unctions is more diicult than computing the it itsel with other methods! We shall not see or use this theorem much, i at all, so I will not give you any eamples yet. The ormal proo o this theorem is also rather technical, requiring the ormal, technical deinition o a it, but its content is very intuitive and demonstrated by this picture: g c g c I trust that the picture well illustrate the reason why the name o Squeeze Theorem is used. And you may also see it reerred to as the Sandwich Theorem, especially by people like me who enjoy ood! Summary Common sense principles can be used to determine a it when the quantities involved can be related in a deinite way. Limits o this kind can be computed ormally, but using logic can provide a aster and equally sae method. But still use with care! Common errors to avoid Be careul in the use o the three laws presented in this section and avoid hidden issues that may make them invalid. Eamples o such issues will be seen later. Dierential Calculus Chapter : Limits and continuity Section : Basic properties o its Page 5
Learning questions or Section D - Review questions:. Describe in your own words, but accurately, the concept I call Anti-Murphy s Law.. Eplain what it means or a unction to have an ininite it.. Describe in your own words, but accurately, the concept I call Law o balloons. 4. Describe in your own words, but accurately, the concept I call Law o gravity. 6. Eplain why i c depending on the sign. and g 0, then c 7. Eplain why i is bounded as c and 0 c g. c g c g, then, 5. Describe in your own words, but accurately, the Squeeze theorem. Memory questions:. What is the value o. What is the value o. What is the value o 4. What is the value o? 0? 0? 0 ln? 0 5. When is it true that ( ) g( ) ( ) g( )? a a a 6. When is it true that ( ) g( ) ( ) g( )? 7. When is it true that a a a ( ) ( ) a a g ( ) g ( )? a Dierential Calculus Chapter : Limits and continuity Section : Basic properties o its Page 6
Computation questions: Use any o the laws presented in this section to compute the its presented in questions -4.. 0. 7 6 0. ln 0.. 4. 8 5 4 5. 4 4 6. 7. 8. 4 4 4z 9. z z 4... 4. 0 0 6 69 69 6 e 5. 0 6. 0 e 7. 8. 9. e 0 / e 0 / 6 ln 4. / Dierential Calculus Chapter : Limits and continuity Section : Basic properties o its Page 7.. 4. 5. 6. 7. 8. 9. 0 cos cos sin cos cos cos sin sin sin sin 0 sin
0. cos. tan 7 sin 4. 4. sin ln. 0 Theory questions:. Is it true that i k is a real number and, then h a? k kh a Proo questions:. Determine whether sin cos eists or not. I it does, ind it. Templated questions:. Try applying the methods discussed in this section to any it you may need or see. What questions do you have or your instructor? Dierential Calculus Chapter : Limits and continuity Section : Basic properties o its Page 8