<Text-field style="_cstyle284" layout="Heading 1"><Font bold="true">Take It to the Limit</Font></Text-field> Note: You may notice differences between this Maple worksheet and the equivalent Mathematica notebook. These differences were introduced to preserve the content of these modules and were necessary because of major functional differences between Maple and Mathematica. In executing this module, you can expect to see an error in the evaluation. This will happen whenever you try to divide by 0, as we do in certain evaluations.
<Text-field style="Heading 1" layout="Heading 1">Introduction</Text-field> OBJECTIVE: To interpret the limit concept graphically and numerically. In this module, you will explore some limits by graphing the functions involved and by creating tables of values for the functions. You will explore, in some detail, functions that have indeterminate forms at the limit point, and you will come to appreciate more fully the importance of formal proofs in mathematics.
<Text-field style="_cstyle290" layout="Heading 1"><Font size="18">Technology Guidelines</Font></Text-field> NOTE: If you have just finished a worksheet, restart Maple before executing a new worksheet. TO OPEN SECTIONS, Click on the Arrow at the left hand side of the screen or select Expand All Sections from the View drop down menu. TO STOP AN EXECUTION Click on STOP button from the toolbar. ORDER OF EXECUTION Execute commands in the order given. Do not skip any Maple Input lines within a given worksheet Alternatively, you can execute the entire worksheet by selecting the Execute Worksheet command from the Edit drop down menu. SAVING WORKSHEETS. You can save anytime to any directory you choose, and it is wise to save often. EXPERIENCING MAJOR PROBLEMS Save if appropriate and then shut down Maple and start it up again.
<Text-field style="Heading 1" layout="Heading 1">Part I: What is 0/0 ? </Text-field>
<Text-field style="Heading 3" layout="Heading 3"></Text-field> When we try to evaluate a function like 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 at x=0, we get LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1GIzYkLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Y0Ri4vJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRjwvJSliZXZlbGxlZEdRJmZhbHNlRidGNA== , but what does this mean? The answer to the question "What is 0/0?" depends on whether the numerator or the denominator of the rational function we are trying to evaluate approaches 0 faster. This function is not defined at x=0. because its denominator is 0 there; however, the numerator is also 0 at x=0. To better understand the situation, let's investigate what happens to the values of 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 when x is very near 0. We begin by graphing the function. restart: f:=x-> x*(1-cos(x))/(x-sin(x)); plot(f(x), x=-20..20, labels=[x,y]); The graph seems to indicate that this function is approaching 3 as x approaches 0. To see that this is the case, let's compute some values of the function for x near 0. The matrix command provides a way to do this, and we use it to create a list of entries, {x, f(x), f(x)-3} for values ranging from -1 to 1 in increments of 0.1. Don't be surprised by the error messages that are produced when you execute the next command; they are to remind you that f(x) is not defined at x=0. Digits:=8: count:=0: for i from -1.0 to 0 by 0.1 do mm[count]:=([i,f(i), f(i)-3]); count:=count+1: od: matrix([[`x`, `f(x)`, `f(x)-3`], seq(mm[i], i=1..count-1)]); Digits:=8: count:=0: for i from 1.0 to 0 by -0.1 do mm[count]:=([i,f(i), f(i)-3]); count:=count+1: od: matrix([[`x`, `f(x)`, `f(x)-3`], seq(mm[i], i=1..count-1)]); How close to 0 must I hold x in order to guarantee that f(x) stays within 1/100th of 3? You will explore that in the You Try It: Part I section.
<Text-field style="_cstyle286" layout="Bullet Item">Continuity Issues: </Text-field> You might notice from the graph of f(x) that it appears to be continuous. For a function to be continuous, the limit as x approaches 0 must exist and this condition seems to be met. However, continuity requires that the function be defined when x is 0, and here we have a problem. If we evaluate f(0), the result is an indeterminate form (0/0 is such a form). f(0); Is f(x) continuous at x=0? why? The graph of f(x) is deceiving. Mathematically, there is a gaping hole in the graph at the point (0,3) even though we can't see it on the Maple graph of f(x). As a result, we might be falsely led to believe that f(x) is continuous at x=0. Now consider the function, 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 if LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVElJm5lO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4= and g(x) = 3 if x=0. What is g(0)? What is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KC1JJ211bmRlckdGJDYlLUkjbW9HRiQ2LlEkbGltRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR0Y6LyUqc2VwYXJhdG9yR0Y6LyUpc3RyZXRjaHlHRjovJSpzeW1tZXRyaWNHRjovJShsYXJnZW9wR0Y6LyUubW92YWJsZWxpbWl0c0dRJXRydWVGJy8lJ2FjY2VudEdGOi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjE2NjY2NjdlbUYnLUYjNiYtRiw2JFEieEYnL0Y8USdpdGFsaWNGJy1GNTYuUScmcmFycjtGJ0Y4RjtGPkZARkJGREZGL0ZJRjpGS0ZNL0ZRRk8tSSNtbkdGJDYlUSIwRidGOEY7RlgvJSxhY2NlbnR1bmRlckdGOkYrLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjRlbUYnLyUmZGVwdGhHRmRvLyUqbGluZWJyZWFrR1ElYXV0b0YnLUYjNiYtRiw2JFEiZ0YnRlgtRjU2LVEwJkFwcGx5RnVuY3Rpb247RidGO0Y+RkBGQkZERkZGZ25GS0ZNRmhuLUkobWZlbmNlZEdGJDYlLUYjNiRGVUZYRjhGO0ZYRitGWEYrRlg= ? Is this function continuous at x=0? The discontinuity in f(x) at x=0 is said to be "removable" since we can remove it by forming a new continuous function, g(x), that is identical to f(x) when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVElJm5lO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4= and is equal to 3 when x=0. For further exploration of continuous and discontinuous functions, refer to "Continuous and Discontinuous Curves," a JAVA applet included in this supplement.
<Text-field style="Heading 1" layout="Heading 1">You Try It: Part I</Text-field>
<Text-field style="_cstyle287" layout="Bullet Item">Take the Epsilon Challenge</Text-field> We have a challenge for you. Here it is. Determine how close x must be to 0 so that the values of f(x) will be within a specified distance from 3. More specifically, we challenge you with a small positive number, let's call it LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEtJnZhcmVwc2lsb247RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy, and ask you try to determine a value for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJmRlbHRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== so that when 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 < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJmRlbHRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVElJm5lO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4=, the values of f(x) will satisfy the inequality, 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 , that is, the distance between the value of f(x) and 3 will be less than LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEtJnZhcmVwc2lsb247RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy. For example, if LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUS0mdmFyZXBzaWxvbjtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiPUYnRjcvJSZmZW5jZUdGNi8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y2LyUqc3ltbWV0cmljR0Y2LyUobGFyZ2VvcEdGNi8lLm1vdmFibGVsaW1pdHNHRjYvJSdhY2NlbnRHRjYvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZOLUkjbW5HRiQ2JFElMC4wMUYnRjdGN0YrRjc= the table of values in the cell above shows that 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 - LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjNGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSI8RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZNLUYyNiRRJDAuMUYnRjVGNUYrRjU= is not satisfied when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSgmZGVsdGE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0Y3LyUmZmVuY2VHRjYvJSpzZXBhcmF0b3JHRjYvJSlzdHJldGNoeUdGNi8lKnN5bW1ldHJpY0dGNi8lKGxhcmdlb3BHRjYvJS5tb3ZhYmxlbGltaXRzR0Y2LyUnYWNjZW50R0Y2LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiRRIzEuRidGN0Y3RitGNw== You need to find a smaller LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJmRlbHRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== that will keep x closer to 0 so that you can beat our LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEtJnZhcmVwc2lsb247RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy challenge. To help you with this, we include a command in the next cell that generates a list of ordered pairs 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 so that you can look at the magnitude of the differences between the values of f(x) and 3 to see if the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJmRlbHRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== you picked meets our LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEtJnZhcmVwc2lsb247RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy challenge. In addition, we added to each ordered pair the result of a test to see if a is within a distance LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEtJnZhcmVwc2lsb247RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy of 3. We start with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUS0mdmFyZXBzaWxvbjtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiPUYnRjcvJSZmZW5jZUdGNi8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y2LyUqc3ltbWV0cmljR0Y2LyUobGFyZ2VvcEdGNi8lLm1vdmFibGVsaW1pdHNHRjYvJSdhY2NlbnRHRjYvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZOLUkjbW5HRiQ2JFElMC4wMUYnRjdGN0YrRjc= and a value for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJmRlbHRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== that is too big. You should try different values for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJmRlbHRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg==, re-evaluating the commands in the cell each time, until you find a LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJmRlbHRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== that works. After you find a LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJmRlbHRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== that works for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUS0mdmFyZXBzaWxvbjtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiPUYnRjcvJSZmZW5jZUdGNi8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y2LyUqc3ltbWV0cmljR0Y2LyUobGFyZ2VvcEdGNi8lLm1vdmFibGVsaW1pdHNHRjYvJSdhY2NlbnRHRjYvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZOLUkjbW5HRiQ2JFElMC4wMUYnRjdGN0YrRjc=, try to find LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJmRlbHRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg==-values that work for 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, .... Digits:=25: delta:=0.0025: epsilon:=0.01: L:=3: count:=0: for i from -delta to delta by delta/10 do if (evalf(abs(f(i)-L)) < epsilon) then result:=true: else result:=false : fi: mm[count]:=([i, abs(f(i)-L), result]); count:=count+1 od: matrix([[`x` , `|f(x)-L|` , `|f(x)-L|< epsilon`], seq(mm[i], i=1..count-1)]); The formal definition of the limit says that if you are able meet our LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEtJnZhcmVwc2lsb247RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy challenge, no matter how small we make our LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEtJnZhcmVwc2lsb247RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy and if you are able to prove that you can do, then you can say unequivocally that 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. We aren't going to ask you to prove this limit, but your instructor may ask you to prove some simpler ones. One thing you can do, however, is use Maple's limit( ) command to see if it verifies our conjecture that 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. limit( f(x), x = 0);
<Text-field style="_cstyle288" layout="Bullet Item">What's Up? (and Down?)</Text-field> We can get more insight into what is happening to the function 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 as x gets close to 0 by looking at the numerator and denominator of the function. In the following cell, we plot the numerator and denominator as separate functions of x. We set the viewing window with the range in the option for the plot( ) command so that the relation between the values of the numerator and denominator can be easily seen. You can zoom in closer by letting xb and yb get smaller and smaller. Your bounds for y will have to be much, much smaller than your bounds for x; otherwise you will not be able to see your graphs. num:= x*(1-cos(x)): den:=x-sin(x): xb:= 0.1: yb:= 10^(-4): plot({num, den}, x=-xb..xb, y=-yb..yb, color=[blue,red]); Which curve do you think is the graph of the numerator function, and which is the denominator? Why? See if you can differentiate between the two curves as x gets closer to 0. Use the graphs above to find an approximate numeric relation between the value of the numerator and the value of the denominator for x values that are close to 0. Based upon the relationship you find, explain why the value of 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 is not surprising. Try to generate a table with values of x, num, den, and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1GIzYkLUkjbWlHRiQ2JVEkbnVtRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y4USdub3JtYWxGJy1GIzYkLUYxNiVRJGRlbkYnRjRGN0Y6LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZGLyUpYmV2ZWxsZWRHUSZmYWxzZUYnRjo= in each row, for values of x close to 0. Comment on what is happening to the values of num, den and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1GIzYkLUkjbWlHRiQ2JVEkbnVtRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y4USdub3JtYWxGJy1GIzYkLUYxNiVRJGRlbkYnRjRGN0Y6LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZGLyUpYmV2ZWxsZWRHUSZmYWxzZUYnRjo= as x gets closer to 0.
<Text-field style="ParagraphStyle1" layout="Bullet Item"><Font style="_cstyle402">Try another</Font><Font style="_cstyle401"> </Font><Equation executable="false" style="_cstyle404" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1GIzYlLUkjbW5HRiQ2JVEiMEYnLyUlYm9sZEdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnL0Y4USVib2xkRicvJStmb250d2VpZ2h0R0Y7Ri4vJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkMvJSliZXZlbGxlZEdGNkY6Rjw=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1GIzYlLUkjbW5HRiQ2JVEiMEYnLyUlYm9sZEdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnL0Y4USVib2xkRicvJStmb250d2VpZ2h0R0Y7Ri4vJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkMvJSliZXZlbGxlZEdGNkY6Rjw=</Equation><Font style="_cstyle405"> </Font><Font style="_cstyle403">expression</Font></Text-field> Find the limit as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEnJnJhcnI7RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSSNtbkdGJDYkUSIxRidGPkY+RitGPg== of the function 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 Start by trying to evaluate the function at x =1. f:=x->log(x)/(1-2^(x-1)); f(1); We had better graph the function to see what is happening. plot(f(x), x=0.5..1.5, labels=["x","y"]); How can you determine the precise limit? Try looking at some values of the function for x near 1. count:=0: for i from 0.9 to 1.1 by 0.01 do mm[count]:=([i, (f(i))]); count:=count+1: od: matrix([[`x`, `f(x)`], seq(mm[i], i=1..count-1)]); To investigate this limit, try some explorations like those in Part I.
<Text-field style="Heading 1" layout="Heading 1"><Font style="_cstyle278" size="18">Part II: What is </Font><Equation executable="false" style="2D Comment" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJW1zdXBHRiQ2JS1JI21uR0YkNiRRIjBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGNC8lMXN1cGVyc2NyaXB0c2hpZnRHRjdGOEYrRjg=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJW1zdXBHRiQ2JS1JI21uR0YkNiRRIjBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGNC8lMXN1cGVyc2NyaXB0c2hpZnRHRjdGOEYrRjg=</Equation><Font style="_cstyle279" size="18"> ?</Font></Text-field> How does a function like LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkobWZlbmNlZEdGJDYoLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GOVEnbm9ybWFsRicvSSttc2VtYW50aWNzR0YkUSRhYnNGJy8lJW9wZW5HUSkmdmVyYmFyO0YnLyUmY2xvc2VHRkJGPUYxLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y7 behave as x gets closer and closer to 0? The answer is not obvious, since numbers to the 0 power got to 1, but 0 to any power is 0. We will start by trying the limit command. restart; g:=x->abs(x)^x; limit(g(x), x=0); Is this correct? We might look at the graph of the funciton. Note what happens when x is 0 on the graph. plot(g(x), x=-2..2, labels=[x,"g(x)"]);
<Text-field style="_cstyle289" layout="Bullet Item">Continuity Issues</Text-field> If we ask for g(0), the result should indicate that it is an indeterminate form (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJW1zdXBHRiQ2JS1JI21uR0YkNiRRIjBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGNC8lMXN1cGVyc2NyaXB0c2hpZnRHRjdGOEYrRjg= is such a form). g(0); The graph of g(x) appears to be continuous at x=0. Is it? Why?
<Text-field style="Heading 1" layout="Heading 1">Part III: One sided Limits</Text-field> Consider the function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1GIzYmLUkmbXNxcnRHRiQ2Iy1GIzYmLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJ0Y5LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSSNtaUdGJDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvRjpRJ2l0YWxpY0YnRjlGPC1JKG1mZW5jZWRHRiQ2JC1GIzYmRlQtRj02LVEoJm1pbnVzO0YnRjlGQEZDRkVGR0ZJRktGTS9GUFEsMC4yMjIyMjIyZW1GJy9GU0Zgby1GNjYkUSIxRidGOUY5RjlGOS1GIzYkLUZobjYoRmpuRjkvSSttc2VtYW50aWNzR0YkUSRhYnNGJy8lJW9wZW5HUSkmdmVyYmFyO0YnLyUmY2xvc2VHRl5wRmlvRjkvJS5saW5ldGhpY2tuZXNzR0Zkby8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZlcC8lKWJldmVsbGVkR0ZCRjk= . This function appears to have problems at x = 1. Let's begin by looking at the graph. h:=x-> sqrt(2*x)*(x-1)/abs(x-1): plot(h(x), x=0..3, labels=[x,y]); The vertical line at x = 1 really shouldn't be there. You can remove the line by setting the print options discont=true. The lower piece is not really connected to the upper piece. Maple graphs a function by plotting a series of points on the funciton and then connects them with straight lines, sometimes when they really shouldn't be connected. What is happening at x = 1? is h(x) defined at x = 1? h(1); Lets' try the limit() command. Becasue of the input at x=1, we will use on-sided limits. The option left indicates that we are approaching the value specified from values that are smaller , whereas right indicates that we are approaching the value specified from values that are larger. limit(sqrt(2*x)*(x-1)/(-(x-1)), x=1, left); limit(sqrt(2*x)*(x-1)/((x-1)), x=1, right); This result verifies what is shown in the graph. That is 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 and 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 . We can check this out by looking at some values of the function x = 1 and comparing these values with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbXNxcnRHRiQ2Iy1JI21uR0YkNiRRIjJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== . count:=1: for i from 0.999 to 1.001 by 0.0001 do mm[count]:=([i, (h(i))]); count:=count+1: od: matrix([[`x`, `h(x)`], seq(mm[i], i=1..count-1)]); The value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbXNxcnRHRiQ2Iy1JI21uR0YkNiRRIjJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== is 1.41421
<Text-field style="Heading 1" layout="Heading 1">You Try It: Part III</Text-field> Find the limit as x -> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21pR0YkNiVRJGArYEYnLyUnaXRhbGljR1EldHJ1ZUYnL0YzUSdpdGFsaWNGJy8lMXN1cGVyc2NyaXB0c2hpZnRHRjFGMg== of 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. g:=x->x*log(abs(x)): g(0); plot(g(x), x=-1..2); What do you think is happening around x = 0? How do we know for sure that there is not a jump discontinuity at x = 0? Maybe, if we zoom in close enough to x = 0, we will see a jump. How can we tell for sure whether ther is or is not a jump at x = 0? limit(x*log(x), x=1, right);
<Text-field style="Heading 1" layout="Heading 1">Part IV: What a Difference a Power Makes!</Text-field> In your coursework, you have undoubtedly studied the limit of the 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 as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEnJnJhcnI7RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSSNtbkdGJDYkUSIwRidGPkY+RitGPg== and have found that its value is 1. What happens to the limit if we change the power of x in the denominator to values that are near but not equal to 1? We begin by graphing three of these functions and see what the graphs suggest. The function 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 see what the graphs suggest. The function 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 will be graphed in red, 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 in green, and 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 in blue. Because the roots of negative numbers are not always real numbers, we use a right-hand limit, approaching 0 only from the right. restart: f1:=sin(x)/x: f2:=sin(x)/x^(0.9): f3:=sin(x)/x^(1.1): plot({f1, f2, f3}, x=0..5, color=[red,green, blue]); Based on this graph, what would you guess are the limits of the three functions? Now let's evaluate each of the functions at 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 . evalf(subs(x=10^(-100), f1)); evalf(subs(x=10^(-100), f2)); evalf(subs(x=10^(-100), f3)); Do these results confirm or contradict the conjectures you made for the values of the three limits?
<Text-field style="Heading 1" layout="Heading 1">You Try It: Part IV</Text-field> Proofs are Important! Closely examine the 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 problem. The following commands will result in the sine functinon being plotted in the thicker form, x in red, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJW1zdXBHRiQ2JS1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW5HRiQ2JFEiOUYnL0Y7USdub3JtYWxGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGQUYrRkE= in green and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRJDEuMUYnL0Y2USdub3JtYWxGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGPA== in blue. num:=sin(x): den1:=x: den2:=x^(.9): den3:=x^(1.1): b:=10^(-5): plot({num, den1, den2, den3}, x=0..b , y=0..b, thickness=[3,1,1,1], color=[red, green, blue]); LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= What's going on here? Without being more careful, we might conclude from the graphs above that in all three cases the ratio of num to denom is a constant and that each limit should be approaching some finite number. To see what is really happening, you need to look at several graphs of the numerator and denominator functions, each one zoomed by a factor of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1GIzYkLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Y0LUYjNiQtRjE2JFEkMTAwRidGNEY0LyUubGluZXRoaWNrbmVzc0dGMy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZALyUpYmV2ZWxsZWRHUSZmYWxzZUYnRjQ= . Go back to the preceding cell and re-execute the commands with the exponent of b = -7, -9, -11, . . . What does this tell you about the ratio of num to denom in each of the three limits as x approaches 0? Wait a minute! How do we know for sure that the pattern illustrated in the graphs above will continue if we zoom in even closer. Maybe thins will "settle down" and the ratios will, in fact, approach constant values. What about the function 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 from Part I? Maybe, if we zoom in close enough to 0, the graph of num to denom will start to look like the correspondin graphs for one of the three functions considered in this Part. How can we be sure of the situation? The answer is in the formal definition of the limit and proof. As mathematicians, we are not fully satisfied until we can prove our conjectures about the behavior of the functions as x approaches 0. Then, there is no doubt.