1.1 Limits: Graphically and Numer ically1.1 Limits:Graphically and NumericallyThere are three ways to calculate the limit of a function: graphically, numerically, and algebraically. This sectionfocuses on the first two. We will introduce limits using the graphical approach to get an intuitive idea for what alimit is.Determining Limits GraphicallyConsider the graph of f :11 22 33 441122334400What is f (2)?What y-value does f (x) approach when x is getting closer to 2 from the left? How do we write this usingmathematical notation?What y-value does f (x) approach when x is getting closer to 2 from the right? How do we write this usingmathematical notation?Whaty-value doesf(x) approach whenxis getting closer to 2? How do we write this using mathematical notation?!Notice that the value of the function is not necessarily equal to the limit!© TAMU  Companion to Calculus for Business and Social Sciences, by Allen, A. and Orchard, P. (2021)1io÷.o¥:*etataf(2)=3'¥:-¥m2_f"-marinationnotation÷:{I¥I¥'✗calvesaregreaterthan2__:a-¥fc✗1.1 Limits: Graphically and Numer icallyThe previous discussion leads to the following:Theorem 1.1 For real numbers c and L, limx!cf (x) = L if and only if limx!cf (x) = limx!c+f (x) = L.⌅ Example 1 Using the graph of f shown below, estimate the following limits.-4-4 -3-3 -2-2 -1-1 11 22 33 44 55 66 77 88 99 1010 1111 1212-2-2-1-1112233445566778800a. limx!9f (x)b. limx!9+f (x)c. limx!9f (x)d. limx!3f (x)e. limx!2+f (x)f. limx!6f (x)⌅The previous example demonstrates an important consequence of Theorem 1.1:Theorem 1.2 For some real number c, if limx!cf (x) , limx!c+f (x), then limx!cf (x) does not exist.Determining Limits NumericallyWhen estimating limits numerically, we analyze they-values of the function asxgets "closer" to a specifiedx-valuefrom both the left and right. In other words, we substitute x-values into the function and look for a trend in they-values.⌅ Example 2 Estimate limx!2x2+ 4x 12x 2numerically, if it exists.⌅© TAMU  Companion to Calculus for Business and Social Sciences, by Allen, A. and Orchard, P. (2021)2ygCDNE)4$CDNE)*⑥&--8=5g)=same=0=8£☐↳WEE.÷:¥÷¥÷*→1.1 Limits: Graphically and Numer ically⌅ Example 3 Estimate limx!53(x + 5)2numerically, if it exists.⌅We can describe the way the limit in the previous example does not exist by writinglimx!53(x + 5)2!1and limx!5+3(x + 5)2!1Because the function is approaching positive infinity from both the left and right, we can write one limit:limx!53(x + 5)2!1!This limit does not exist! Using1or1merely describes the way in which the limit does not exist. We wantas much information as possible, so in future examples we will also describe such infinite behavior whereapplicable.This leads us to discussing infinite limits (i.e., limits where the function approaches positive or negative infinity).Infinite LimitsIn the previous example, the function had what is called a vertical asymptote at x = 5. More precisely:DefinitionLetfbe a function. If any of the following conditions hold for some real numberc, then the linex=cis avertical asymptote of the graph of f :limx!cf (x) !1 or limx!cf (x) ! 1limx!c+f (x) !1 or limx!c+f (x) ! 1limx!cf (x) !1 or limx!cf (x) ! 1⌅© TAMU  Companion to Calculus for Business and Social Sciences, by Allen, A. and Orchard, P. (2021)3Calainstntins1.enterfix)iny=2.VARSSy-vars3.4,(#)1.1 Limits: Graphically and Numer ically⌅ Example 4 Estimate limx!71x 7(a) numerically and (b) graphically. If the limit does not exist, state so and uselimit notation to describe any infinite behavior.⌅� Videos on Tips for Using TI-84 CalculatorVideo #1: Storing functions in Y1Video #2: Storing x-values using STOVideo #3: Adjusting calculator window to view functions© TAMU  Companion to Calculus for Business and Social Sciences, by Allen, A. and Orchard, P.