MA123, Chapter 3: The idea of limits (pp. 47-67, Gootman)Chapter Goals:• Evaluate limits.• Evaluate one-sided limits.• Understand the concepts of continuity and differentiability and their relationship.Assignments: Assignment 04 Assignment 05Earlier, the idea of limits came up naturally in the course of defining the derivative of a function at a point.We now study limits more systematically. Computing a limit means computing what happens to the value of afunction as the variable in the expression gets closer and closer to (but does not equal) a particular value.◮ The basic definition of limit:Let f be a function of x. The expressionlimx→cf(x) = Lmeans that as x gets closer and closer to c, through values both smaller and larger than c, but not equal to c,then the values of f(x) get closer and closer to the value L.Note:It may sometimes happen that the limit does not exist.Example 1 (a):Use the tables to help evaluate limx→2x2+ 8x + 2.x gets close to 2 from the leftx 1.8 1.9 1.99 1.999x2+ 8x + 2x gets close to 2 from the right2.001 2.01 2.1 2.2 xx2+ 8x + 2Example 1 (b): Suppose that, instead of calculating all the values in the above tables, you simplysubstitute the value x = 2 intox2+ 8x + 2. What do you find?Note:The method of substituting in the limiting value of the variable works because the operations ofarithmetic, namely, addition, subtraction, multiplication, and division, all behave reasonably with respect tothe idea of ‘getting closer to’ as long as nothing illegal happens. T he one illegality you will mainly have towatch out for is ‘division by zero’. More precisely, if f and g are two functions one has:limx→cf(x) + g(x)= limx→cf(x) + limx→cg(x) limx→cf(x) − g(x)= limx→cf(x) − limx→cg(x)limx→cf(x) ·g(x)=limx→cf(x)·limx→cg(x)limx→cf(x)g(x)=limx→cf(x)limx→cg(x)as long as limx→cg(x) 6= 021Example 2: Compute limx→1(x2+ 4x + 3) · (2x − 4).Example 3:Compute limx→1x2− 2x + 1x + 1.Example 4:Suppose limx→3f(x) = −2 and limx→3g(x) = 4. Determinelimx→3(x + 1) ·f(x)2+x + 2g(x).◮ Some complications with the definition of limits:The previous examples seem to imply that “com-puting a limit” is the same thing as “evaluating a function”. This is only true if the function in the limit is“nice enough” (“nice enough” will be defined more precisely in a few pages).The next few examples will illustrate that the computation of limx→cf(x) d oes n ot always reduce to the meresubstitution of the value of c in place of x in the expression defining f(x). The ‘unusual’ functions described inwhat follow s are introduced to emphasize the fact that the notion of limit really involves what happens to thevalues of f (x) as x gets closer to th e fixed value c, and not what the value of f (x) at x = c is. In addition, themost interesting limits generally arise p recisely when substitution gives an illegal expression involving divisionby 0, or even an expression of the form00. The latter case occurs for example when computing the derivativeof a function.◮ How can a limit fail to exist?There are two basic ways that a limit can fail to exist.(a) The function attempts to approachmu ltiple values as x → c.Geometrically, this behavior can be seen as a jump in the graph of a function.Algebraically, this behavior typically arises with piecewise defined funtions.(b) The function grows withou t bound as x → c.Geometrically, this behav ior can be seen as a vertical asymptote in the graph of a function.Algebraically, this behavior typically arises when the denominator of a function approaches zero.22Example 5:The graph of the fun ctionh(x) =(x2− 3, if x > −2;2x + 7, if x ≤ −2is shown to the right.Analyze limx→−2h(x).xy−213The p revious example showed that the limit of a h(x) as the variable approached −2 did not exist. On theother hand, the function appears to have well defined limiting behavior on either s ide of x = −2. This bringsus to the following notion s:One-sided limits:A one-sided limit ex presses what happens to the values of an expression as the variablein the expression gets closer an d closer to some particular value c from either th e left on the number line (thatis, through values less than c) or from the right on the number line (that is, th rough values greater than c).The notation is:limx→c−f(x)|{z }limit from the left of climx→c+f(x)| {z }limit from the right of cFact:limx→cf(x) exists if and only if both limx→c−f(x) and limx→c+f(x) exist and have the same value.Example 6:The graph of the fun ctiong(x) =4xx2+ 1, if x 6= 1;3, if x = 1.is shown to the right.Compute limx→1g(x).x g(x)0.8 1.95121950.91.98895030.9991.9999991.001 1.9999991.11.99095021.21.9672131xy12323◮ The problem of division by zero and a finite nonzero numerator: When this happens, it is stan-dard to say that the expression “is getting arbitrarily large (in the positive or negative direction)” or is “goingto (positive or negative) infinity,” denoted by ±∞. As infinity is n ot really a number, the expression is notreally getting close to any particular real number. Thus, technically speaking, the limit does not exist. In theweb homework system, “infinite limits” should be entered as “DNE”.Example 7:Analyze limx→15(x − 1)2.Example 8:Analyze limx→12x − 1.Example 9:Analyze the limit limx→02√xBe sure to graph the functions in each of the last three examp les, and notice the graphs have vertical asymptotesat x = 1, x = 1, and x = 0, respectively.24◮ The case00:The most interesting and important situation with limits is when a substitution yields00. This is p recisely the situation we are confr onted with when attempting to compute derivatives f rom thedefinition. The result00yields absolutely no information about the limit. It does not even tell us that th e limitdoes n ot exist. The only thing it tells us is that we have to do more work to determine the limit.Example 10:Find the limit limx→04xx.Example 11:Find the limit limx→02x+5x − 2x.Example 12:Find the limit limx→3x2− 2x − 3x − 3.xy34Example 13: Find the limit limh→0(h − 3)2− 9h.25Example 14: Find the limitslimx→2+|3x − 6|x − 2limx→2−|3x − 6|x − 2limx→2|3x − 6|x − 2.◮ Limits at infinity:A function f(x) is said to be a rational function if it is of the typep(x)q(x), wherep(x) and q(x) are both polynomials in x. Sometimes