<Text-field style="_cstyle268" layout="Heading 1">Limits -- "sufficiently close"</Text-field>

Lab 2 -- Limits

The objective of this lab is for you to reach a good understanding of the precise meanings of various types of limits.

Working with your partner: determine which of you will be purple, and which blue. While you must both work through the entire lab, the "purple" person should answer questions/ provide explanations to text written in purple, and the "blue" person should address the blue questions/ explanations. When doing so, the other partner should gently prod for complete and accurate explanations.

Expect to spend the remainder of today's class and possibly another hour working through this file thoroughly. (The goal is understanding -- there are some conceptually challenging ideas here.) You will then likely need another 1 - 2 hours to do the homework.

<Text-field style="Heading 2" layout="Heading 2">Infinite Limits (review and summary of in-class discussion)</Text-field>

Take a look at Definition 4 on p. 99 of your book. (Open your book and read it!) We'll explore what we mean by "values of f(x) can be made arbitrarily large."

Consider 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.

As x gets closer and closer to 0, the values of 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 get bigger and bigger. What is 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? Check out the graph:

plot(1/x^2,x=-1..1,y=0..100);

Another way of phrasing "arbitrarily large" is "as large as we please". Let's explore that idea.

Suppose I asked you to find values of x sufficiently close to 0 so that, when these values for x are plugged into NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNiUtSSNtaUdGJTY5USFGKC8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYoLyUlc2l6ZUdRIzEyRigvJSVib2xkR1EmZmFsc2VGKC8lJ2l0YWxpY0dRJXRydWVGKC8lKnVuZGVybGluZUdGOC8lKnN1YnNjcmlwdEdGOC8lLHN1cGVyc2NyaXB0R0Y4LyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GKC8lK2JhY2tncm91bmRHRkQvJSdvcGFxdWVHRjgvJStleGVjdXRhYmxlR0Y4LyUpcmVhZG9ubHlHRjgvJSljb21wb3NlZEdGOC8lKmNvbnZlcnRlZEdGOC8lK2ltc2VsZWN0ZWRHRjgvJSxwbGFjZWhvbGRlckdGOC8lMGZvbnRfc3R5bGVfbmFtZUdRKzJEfkNvbW1lbnRGKC8lKm1hdGhjb2xvckdGRC8lL21hdGhiYWNrZ3JvdW5kR0ZELyUrZm9udGZhbWlseUdGMi8lLG1hdGh2YXJpYW50R1EnaXRhbGljRigvJSltYXRoc2l6ZUdGNS1JJm1mcmFjR0YlNiotRiQ2Iy1JI21uR0YlNjlRIjFGKEYwRjNGNi9GOkY4RjxGPkZARkJGRUZHRklGS0ZNRk9GUUZTRlVGWEZaRmZuL0ZpblEnbm9ybWFsRihGW28tRiQ2JUYsLUYkNiVGLC1JJW1zdXBHRiU2JS1GLTY5USJ4RihGMEYzRjZGOUY8Rj5GQEZCRkVGR0ZJRktGTUZPRlFGU0ZVRlhGWkZmbkZobkZbby1GY282OVEiMkYoRjBGM0Y2RmZvRjxGPkZARkJGRUZHRklGS0ZNRk9GUUZTRlVGWEZaRmZuRmdvRltvLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGKEYsRiwvJS5saW5ldGhpY2tuZXNzR1EiMUYoLyUrZGVub21hbGlnbkdRJ2NlbnRlckYoLyUpbnVtYWxpZ25HRl5xLyUpYmV2ZWxsZWRHRjgvJStmb3JlZ3JvdW5kR0ZELyUrYmFja2dyb3VuZEdGREYsNyM2IyomIiIiRmpxKiQpJSJ4RyIiI0ZqcSEiIg==, the output is greater than 100.

Interpret the command below. What will the output mean?

solve(1/x^2=100);

This tells us that if x is between 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 and 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, then NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNiUtSSNtaUdGJTY5USFGKC8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYoLyUlc2l6ZUdRIzEyRigvJSVib2xkR1EmZmFsc2VGKC8lJ2l0YWxpY0dRJXRydWVGKC8lKnVuZGVybGluZUdGOC8lKnN1YnNjcmlwdEdGOC8lLHN1cGVyc2NyaXB0R0Y4LyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GKC8lK2JhY2tncm91bmRHRkQvJSdvcGFxdWVHRjgvJStleGVjdXRhYmxlR0Y4LyUpcmVhZG9ubHlHRjgvJSljb21wb3NlZEdGOC8lKmNvbnZlcnRlZEdGOC8lK2ltc2VsZWN0ZWRHRjgvJSxwbGFjZWhvbGRlckdGOC8lMGZvbnRfc3R5bGVfbmFtZUdRKzJEfkNvbW1lbnRGKC8lKm1hdGhjb2xvckdGRC8lL21hdGhiYWNrZ3JvdW5kR0ZELyUrZm9udGZhbWlseUdGMi8lLG1hdGh2YXJpYW50R1EnaXRhbGljRigvJSltYXRoc2l6ZUdGNS1JJm1mcmFjR0YlNiotRiQ2Iy1JI21uR0YlNjlRIjFGKEYwRjNGNi9GOkY4RjxGPkZARkJGRUZHRklGS0ZNRk9GUUZTRlVGWEZaRmZuL0ZpblEnbm9ybWFsRihGW28tRiQ2JUYsLUYkNiVGLC1JJW1zdXBHRiU2JS1GLTY5USJ4RihGMEYzRjZGOUY8Rj5GQEZCRkVGR0ZJRktGTUZPRlFGU0ZVRlhGWkZmbkZobkZbby1GY282OVEiMkYoRjBGM0Y2RmZvRjxGPkZARkJGRUZHRklGS0ZNRk9GUUZTRlVGWEZaRmZuRmdvRltvLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGKEYsRiwvJS5saW5ldGhpY2tuZXNzR1EiMUYoLyUrZGVub21hbGlnbkdRJ2NlbnRlckYoLyUpbnVtYWxpZ25HRl5xLyUpYmV2ZWxsZWRHRjgvJStmb3JlZ3JvdW5kR0ZELyUrYmFja2dyb3VuZEdGREYsNyM2IyomIiIiRmpxKiQpJSJ4RyIiI0ZqcSEiIg== is greater than 100. Another way to write this is to say that

if 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 then 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 > 100.

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bSQiI10hIiM7JCEjXUZpXG0kIiM6IiIiLSUrUFJPSkVDVElPTkc2I0ZnXG0tJSpHUklEU1RZTEVHNiMlLFJFQ1RBTkdVTEFSR0ZhXW0tJSxPUklFTlRBVElPTkc2JCQiIyEpRi0kIjIzKysrKysrNSVGX3ItRmRcbTYjJSVOT05FRy0lKkxJTkVTVFlMRUc2I0Yt

The graph illustrates this: if 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 then x is within the black lines on the graph. For the x-values within the black lines, we see that the output values are greater than 100 (above the blue line).

Your turn (jointly): find a value for a so that if 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 then 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 > 100,000. (What is the appropriate equation to solve? Pencil and paper might help.)

<Text-field style="Heading 2" layout="Heading 2">(Regular) Limits (review and summary of in-class discussion)</Text-field>

The definition on p. 93 uses similar language. (Read it again now.) If the limit of f(x), as x approaches a, is L, we're supposed to be able to make f(x) as close to L as we like, by taking x sufficiently close to a. Again, we'll examine how close is sufficiently close?

Consider the limit: .

f:=x->2*(x^2+x-2)/(x-1);

plot(f(x),x=0..2);

limit(f(x),x=1);

Explain the command above to your partner. What did you just learn?

The limit (L) is 6. Well, what if sufficiently close means within 0.01? How close must we take x to 1? Let's look at the graph more closely.

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

The dotted blue line shows that 0.99 is not sufficiently close to 1: How does f(0.99) illustrate this? (Hint: f(0.99) is more than 0.01 away from 6.)

(Examine the y-values.) But we can see that if x is between roughly 0.995 and 1.005 then the y-values stay between 5.99 and 6.01. (True? False? Discuss!) Let's look more closely.

plot(f(x),x=0.995..1.005);

Bingo. Now the output values are between 5.99 and 6.01. Conclude: if we want f(x) to be within 0.01 of 6, then x must be within 0.005 of 1.

Things get a little messier if the function is not symmetric around the point at which the limit is being taken.

Suppose we want to find a value of c for which values of x that are within c units of 2 give values of NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNiUtSSNtaUdGJTY5USFGKC8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYoLyUlc2l6ZUdRIzEyRigvJSVib2xkR1EmZmFsc2VGKC8lJ2l0YWxpY0dRJXRydWVGKC8lKnVuZGVybGluZUdGOC8lKnN1YnNjcmlwdEdGOC8lLHN1cGVyc2NyaXB0R0Y4LyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GKC8lK2JhY2tncm91bmRHRkQvJSdvcGFxdWVHRjgvJStleGVjdXRhYmxlR0Y4LyUpcmVhZG9ubHlHRjgvJSljb21wb3NlZEdGOC8lKmNvbnZlcnRlZEdGOC8lK2ltc2VsZWN0ZWRHRjgvJSxwbGFjZWhvbGRlckdGOC8lMGZvbnRfc3R5bGVfbmFtZUdRKzJEfkNvbW1lbnRGKC8lKm1hdGhjb2xvckdGRC8lL21hdGhiYWNrZ3JvdW5kR0ZELyUrZm9udGZhbWlseUdGMi8lLG1hdGh2YXJpYW50R1EnaXRhbGljRigvJSltYXRoc2l6ZUdGNS1GJDYlRiwtSSVtc3VwR0YlNiUtRi02OVEieEYoRjBGM0Y2RjlGPEY+RkBGQkZFRkdGSUZLRk1GT0ZRRlNGVUZYRlpGZm5GaG5GW28tSSNtbkdGJTY5USIyRihGMEYzRjYvRjpGOEY8Rj5GQEZCRkVGR0ZJRktGTUZPRlFGU0ZVRlhGWkZmbi9GaW5RJ25vcm1hbEYoRltvLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGKEYsRiw3IzYjKiQpJSJ4RyIiIyIiIg== which are within 0.1 of 4. Remember, the distance between x and 2 is given by 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, and the distance between 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 and 4 is given by 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.

So, we want to find a value for c which makes true the statement:

if 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 then NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNiUtSSNtaUdGJTY5USFGKC8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYoLyUlc2l6ZUdRIzEyRigvJSVib2xkR1EmZmFsc2VGKC8lJ2l0YWxpY0dRJXRydWVGKC8lKnVuZGVybGluZUdGOC8lKnN1YnNjcmlwdEdGOC8lLHN1cGVyc2NyaXB0R0Y4LyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GKC8lK2JhY2tncm91bmRHRkQvJSdvcGFxdWVHRjgvJStleGVjdXRhYmxlR0Y4LyUpcmVhZG9ubHlHRjgvJSljb21wb3NlZEdGOC8lKmNvbnZlcnRlZEdGOC8lK2ltc2VsZWN0ZWRHRjgvJSxwbGFjZWhvbGRlckdGOC8lMGZvbnRfc3R5bGVfbmFtZUdRKzJEfkNvbW1lbnRGKC8lKm1hdGhjb2xvckdGRC8lL21hdGhiYWNrZ3JvdW5kR0ZELyUrZm9udGZhbWlseUdGMi8lLG1hdGh2YXJpYW50R1EnaXRhbGljRigvJSltYXRoc2l6ZUdGNS1GJDYmRiwtRiQ2JS1JI21vR0YlNjNRKSZ2ZXJiYXI7RigvJSVmb3JtR1EmaW5maXhGKC8lJmZlbmNlR0Y4LyUqc2VwYXJhdG9yR0Y4LyUnbHNwYWNlR1EkMGVtRigvJSdyc3BhY2VHRl5wLyUpc3RyZXRjaHlHRjsvJSpzeW1tZXRyaWNHRjgvJShtYXhzaXplR1EpaW5maW5pdHlGKC8lKG1pbnNpemVHUSIxRigvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lMGZvbnRfc3R5bGVfbmFtZUdGVy8lJXNpemVHRjUvJStmb3JlZ3JvdW5kR0ZELyUrYmFja2dyb3VuZEdGRC1GJDYmRiwtRiQ2JUYsLUklbXN1cEdGJTYlLUYtNjlRInhGKEYwRjNGNkY5RjxGPkZARkJGRUZHRklGS0ZNRk9GUUZTRlVGWEZaRmZuRmhuRltvLUkjbW5HRiU2OVEiMkYoRjBGM0Y2L0Y6RjhGPEY+RkBGQkZFRkdGSUZLRk1GT0ZRRlNGVUZYRlpGZm4vRmluUSdub3JtYWxGKEZbby8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRihGLC1GYm82M1EoJm1pbnVzO0YoRmVvRmhvRmpvL0ZdcFEwbWVkaXVtbWF0aHNwYWNlRigvRmBwRmFzL0ZicEY4RmNwRmVwRmhwRltxRl1xRl9xRmFxRmNxRmVxRmdxLUZkcjY5USI0RihGMEYzRjZGZ3JGPEY+RkBGQkZFRkdGSUZLRk1GT0ZRRlNGVUZYRlpGZm5GaHJGW29GYW8tRmJvNjNRIjxGKEZlb0Zob0Zqby9GXXBRL3RoaWNrbWF0aHNwYWNlRigvRmBwRlt0RmNzRmNwRmVwRmhwRltxRl1xRl9xRmFxRmNxRmVxRmdxLUZkcjY5USQwLjFGKEYwRjNGNkZnckY8Rj5GQEZCRkVGR0ZJRktGTUZPRlFGU0ZVRlhGWkZmbkZockZbb0YsNyM2IzItJSRhYnNHNiMsJiokKSUieEciIiMiIiJGW3UiIiUhIiItJSZGbG9hdEc2JEZbdUZddQ==

Discuss. Tranlate from math to English.

First let's plot the function around x = 2 and y = 4.

plot(x^2,x=1.9..2.1,y=3.9..4.1);

Not good enough -- but it does look like keeping x between 1.98 and 2.02 might work.

plot(x^2,x=1.98..2.02,y=3.9..4.1);

Yes, these values for x keep f(x) within 0.1 of 4. (We see 3.9 < f(x) < 4.1 for all the x-values shown.)

We could do this more carefully. We want to know which x values give 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 = 3.9 and 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 = 4.1: these are the outer limits of the allowable region for x. (Why are these the outer limits of the allowable region for x?

fsolve(x^2=4-1/10); fsolve(x^2=4+1/10);

(Note: fsolve is like solve, but fsolve finds one approximate solution to the equation input.)

This shows that keeping x in the interval (1.975, 2.024) guarantees that 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 will be within 0.1 of 4. Note: rounding 2.024845673 up destroys the guarantee -- we need x-values smaller than this number. (WHY?) So we can complete our mathematical expression:

if 1.975 < x < 2.024, then 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

Your turn (jointly): Let 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. Find values for c and d satisfying the statement:

if c < x < d then 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.

f:=x->x^2/3-2*x+1;

<Text-field style="_cstyle269" layout="Heading 1">Limits at Infinity</Text-field>

Remember working with horizontal asymptotes? Here's another way of thinking about them.

You know that 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 is the number that f(x) is getting closer and closer to when you put in values of x closer and closer to a (but not equal to a). What do you think the following limits should be about?

Think about some functions you're already familiar with. What would you say about

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Does this next function have a vertical asymptote? What about a horizontal asymptote? What would you expect this limit to be?

limit((x+1)/(x-1),x=infinity);

plot({(x+1)/(x-1),1},x=100..1000); (The line is y = 1: observe how close the function output values get.)

plot({(x+1)/(x-1),1},x=1000..10000);

Sure enough, as x gets bigger and bigger, the y-values are getting closer and closer to 1.

<Text-field style="Heading 1" layout="Heading 1">Maple commands introduced in this lab (for reference)</Text-field>

The Help files contain copious information, as well as examples. Choosing "Maple Help" in the Help menu above allows you to type in a command and get full information on it. Summaries are provided below.

<Text-field style="Heading 2" layout="Heading 2">limit</Text-field>

Purpose: calculate the limit of a function (when it exists).

Format: limit(function, x = limiting_value)

You can add the option of "left" or "right" for one-sided limits. Examples below.

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

<Text-field style="Heading 2" layout="Heading 2">fsolve</Text-field>

Purpose: solve an equation numerically (approximately). Note: you can specify an interval on which you want Maple to look for the solution.

Format: fsolve(equation, variable_to_solve_for) or fsolve({set of equations}, {set of variables to solve for})

(Note that solve gives all solutions (when it can); fsolve might give only one.)

<Text-field style="_cstyle270" layout="Heading 1">Lab 2 Homework</Text-field>

Submission Instructions: Obtain the file "Lab 2 -- Homework" from BlackBoard (under Maple labs). You will edit this worksheet, then print it and turn in one copy per lab team before class on Tuesday, Sept. 12. (Do not turn in any Maple output this time!) Be sure to give thorough explanations: use both good mathematics and good writing practices. If you wish, you may write in mathematical expressions by hand. You may want to control the size of the graphs by clicking on them and shrinking them.

Potentially Helpful Hint: Get input cells to work in by clicking on the symbol "[>" to the right of the "T" in the menu above.

1. Plot some graphs of 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 centered around 0. (The first is given for you so as to help you obtain a good picture. By telling Maple that the function is discontinuous we prevent Maple from trying to connect points on either side of the y-axis.)

f:=x->tan(Pi*(x+1/2)); plot(f(x),x=-1..1,y=-10..10,discont=true);

Using the graphs, what do you think are the values of 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, 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, and 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?

Follow the type of reasoning used in Infinite Limits section to find a value for b for which 0 < x < b guarantees that f(x)<-10. Repeat this reasoning two more times, corresponding to 1a, b, and c on your worksheet.

Continue with analogous reasoning (and Maple work) to find a value for c for which c < x < 0 guarantees that f(x)>10. Repeat this reasoning two more times, corresponding to 1d, e, and f on your worksheet.

2. Let 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. (Your responses will go on your worksheet.)

Graph for x between 0.5 and 1.5, then estimate L.

Use Maple's limit command to find L. (The limit command was demonstrated above. Feel free to use the help section, too. You may want to use evalf to get a numerical approximation for L.)

Find values for b and c for which b < x < c guarantees that the difference between 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 and L is less than 0.01. (Hint: 0.1572562364 is not an answer you want!)

3. Let 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. Plot the function and find the limit as x approaches 3, then do the Maple work necessary to complete #3 on the worksheet.

4. Do these functions have horizontal asymptotes? Experiment by evaluating the limits of each function at 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 and 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. Plot the functions that you think do have horizontal asymptotes together with the asymptote(s). (Refer back to the section {Limits at Infinity). Copy and paste your graphs into the worksheet, and provide an explanation of your answer. (You will probably have to generate multiple graphs to produce a graph that illustrates your response well.)

a) 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

b) 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