Lecture for Week 3 Secs 2 5 and 2 6 Infinity and Continuity including vertical asymptotes from Sec 2 2 1 Infinity is not a number It is a figure of speech Stewart pp 86 88 and 109 shows graphical examples of functions with vertical asymptotes or equivalently infinite limits The asymptote is a vertical line which is approached by the graph Typically it appears at a spot where the denominator of the function vanishes equals 0 but the numerator is not zero 2 lim f x x a means that the values of f x can be forced to be arbitrarily large and positive by considering only x sufficiently close to a This definition doesn t allow x to be actually equal to a Usually f a is not even defined 3 Note that a function like 1 x sin x has arbitrarily large values near a 0 but it does not approach because it also has small values arbitrarily close to a For instance it s 0 when x N1 The limit is if the function becomes arbitrarily large and negative around the asymptote 4 Often a function will approach from one side and from the other Exercise 2 2 18 p 90 extended 6 around Discuss the behavior of f x x 5 x 5 5 As x 5 from the right the denominator becomes very small but remains positive 6 lim x 5 x 5 As x 5 from the left the denominator becomes very small and negative lim x 5 6 x 5 The line x 5 is a vertical asymptote 6 In Sec 2 6 we have graphs with horizontal asymptotes representing functions with definite limits at infinity The asymptote is a horizontal line that the graph approaches at either the extreme right of the graph or the extreme left or both Notice last picture on p 123 that the graph does not need to stay on one side of the asymptote it can wiggle around it 7 For example lim f x L x means that the values f x can be forced to be arbitrary close to L by considering only values of x that are sufficiently large and negative More precisely I should say negative and sufficiently large in absolute value 8 Exercise 2 6 33 p 133 x 4 lim x x2 2x 5 Exercise 2 6 17 lim 1 x x x 9 x 4 lim x x2 2x 5 is a rational function ratio of two polynomials The basis trick for finding limits of such things at infinity is Divide numerator and denominator by the highest power appearing in the denominator 10 4 1 x 4 x x2 2 5 2 x 2x 5 1 x x2 The point of this maneuver is that now the denominator approaches 1 as x so all we need to do is to take the limit of the numerator which is 0 in this case x 4 0 lim x x2 2x 5 11 It s easy to see what will happen in all problems of this type 1 If the denominator has higher degree than the numerator the limit is 0 2 If the numerator and denominator have the same degree the limit is some nonzero number the coefficient of the leading term in the numerator 3 If the numerator has the higher degree the 12 two limits at infinity are infinite possibly of opposite signs Example x x3 3 x2 1 1 x3 3 lim x x2 1 3 x2 1 x2 x3 3 lim x x2 1 With rational functions we can t construct an example for which limx and limx are finite and different But the inverse trig function tan 1 x has that property see graph p 279 13 In a problem like lim 1 x x x it helps to rationalize the numerator by multiplying numerator and denominator by the conjugate expression in this case 1 x x You get 1 x x 1 x x The numerator now goes to 1 while the denominator goes to infinity so the limit is 0 14 In both of our exercise examples the horizontal axis y 0 was a horizontal asymptote Here is an example with a different result Exercise 33 extended x and Find any asymptotes of y f x x 4 state the corresponding limits involving infinity 15 There is a vertical asymptote at x 4 lim f x lim f x x 4 x 4 lim f x lim x 1 1 4 x 1 There is a horizontal asymptote at y 1 16 Now what is continuity The intuitive idea is that a function is continuous if its graph can be drawn in one stroke never lifting the pencil from the paper y x 17 The practical meaning of continuity for doing calculations is related to something we talked about last week Remember that the crucial practical question about limits is when do we know that lim f x f a x a More generally in evaluating limits we often wanted to push the limit through a function 18 lim f g u f u b lim g u u b If limu b g u exists and equals a then that maneuver is correct provided that lim f x f a x a If is true we say that f is continuous at a a very convenient property for a function to have 19 If a function fails to be continuous at a then one of three things has happened 1 f a is not defined 2 lim f x does not exist x a 3 Those two numbers exist but are not equal It is possible for both 1 and 2 to happen at once 20 The limits that define derivatives of the type g x g a lim x a x a are the classic example of discontinuous functions of type 1 They are the main reason for studying limits at the beginning of calculus We have seen various examples of type 2 including a vertical asymptotes b points where left and right limits exist but are not equal 21 It is easy to draw graphs of functions of type 3 take a continuous function and move one point on it vertically to a strange place It is harder to find such functions in real life but here s an attempt Suppose you shoot a gun at a target containing a hole exactly the size of a bullet If your aim is perfect the bullet lands on the far side Otherwise it ricochets off the target and lands on your side 22 Another way to summarize continuity is that nearby inputs yield nearby outputs If x is close to a then f x is close to f a As it stands this is rather vague and says nothing about f x for any one particular x The precise definition of a limit and hence of continuity is a statement about the collective behavior of all points x f x for x sufficiently near a 23 Exercise 2 5 23 Explain why h x 5 x 1 x2 2 is continuous on …
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