MATH 148 Fall 2006 Lecture Notes, Friday, 6 October 2006LimitsWhat does “limx!af.x/ D L” mean?What does “The number L is the limit of the function f.x/ as x goes to a” mean?Answer “For any prescribed, given (positive) distance, if x is near enough to a, then the distancebetween f.x/ and the number L is less than the given distance.”So, if the number L is 15 and the given distance is 1=10, then the distance between f.x /and the number 15 is supposed to be less than 0:1,ifx is near enough to a. This means thatf.x/ is between 14:9 and 15:1, namely 14:9 <f.x/<15:1. Written differently this says thatjf.x/ 15j < 0:1.Note that “near enough” means that x is only near a, not actually equal to a, that is, x ¤ a.Now the distance between f.x/ and the number L may be written as jf.x/ Lj. So, if theprescribed, given distance is some number d , then we have that jf.x/ L j < d or L d <f.x/<L C d. That means that f.x/ is between the two numbers L d and L C d, that f.x/lies in the open interval .L d; L C d/.So now what does “x is near enough to a” mean?This just means that the number x is in some (possibly small?) interval around the number a.For example, it might be the case that x is near enough to a when the distance between x and ais less than 3, say. So it might be the case that if x is in the open interval .a 3; a C 3/, then thedistance between f.x/ and the number Lis less than the given distance. Note this means that forany x between a 3 and a C3 that is different from a the distance between f.x/ and the numberL would be less than the given distance. More generally, if ` is the length of the interval around afor which x is near enough to a, then we have 0 < jx aj <`. This says that x is near enoughto a if either a `<x < a or if a < x < a C `. For any such x,ifL is the limit of f.x/ as xgoes to a, then the distance between f.x/ and the number L will be less than the prescribed, givendistance.For one-sided limits, such as limits from the left, we have a slightly different understanding ofwhat “near enough” means. What the one-sided limitlimx!af.x/ D Lis telling us is that the number L is the limit of the values of the function f.x/ when x is near a,but less than a. Here the minus () sign is telling us that x is to the left of (more negative than)the number a.So suppose that all numbers that are within120-th of a are near enough, as long as they are lessthan a. That would mean that x would be near enough if a 0:05 < x < a, that is, if x were in theopen interval .a 0:05; a/. This set is just “half” of that for the “normal” (two-sided) limit notion,namely the left half.Taking limits from the right is handled similarly. The notation is limx!aCf.x/ D L, with theplus (C) sign telling us that we are only interested when x is to the right of a, namely only whena < x.1What about infinite limits?That is, suppose that there is no (finite) real number L that is the limit of f.x/ as x goes to a.What doeslimx!af.x/ D1mean ?Answer For any given (prescribed) number M whatsoever, if x is near enough to a then f.x/is greater than M . So, for example, if the given number is 11 or 1000 or whatever, then f.x/ willbe greater than 11 or 1000 or whatever the number M is, as long as x is near enough to a.Example limx!31.x 3/2D1Remarks(1) The notion of a one-sided limit is handled here in the same way as before, namely the notionof x being near enough is treated in the same way. For example, the one-sided limit from theright is indicated with a plus sign, such as limx!1C1.x 1/D1, telling us that only values ofx greater than 1 are being considered.(2) The notion of minus infinity as the value of a limit, namely, limx!af.x/ D1, just meansthat f.x/ is less than any given number, as long as x is near enough to a. For example,limx!11.x 1/2D1.Some Limit Facts(1) If f.x/ g.x/ for x near a, then limx!af.x/ limx!ag.x/.(2) (Squeeze Theorem)Iff.x/ h.x/ g.x/ for x near a and if L D limx!af.x/ D limx!ag.x/,then limx!ah.x/ D L.(3) (Direct Substitution for Polynomials)IfP.x/ is a polynomial, then limx!aP.x/ D P .a/.(4) (Direct Substitution for Rational Functions)Iff.x/ D P. x/=Q.x/, P and Q are polynomi-als, and a is in the domain of f , that is, if Q.a/ ¤ 0, then limx!af.x/ D f.a/ D P .a/=Q.a/.Two 0/0 Examples(1)x2 4x 2D.x C 2/.x 2/x 2D x C2 implies limx!2x2 4x 2D limx!2x C 2 D 4.(2)px C 5 3x 4Dpx C 5 3x 4 px C 5 C3px C 5 C3D.x C 5/ 9.x 4/.px C 5 C 3/D1px C 5 C 3solimx!4px C 5 3x 4D limx!41px C 5 C3D1p4 C 5 C
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