<Text-field style="_cstyle27" layout="_pstyle17">Calculating Limits</Text-field>

Lab 3 -- Limits, Average and Instantaneous Velocity

The objectives of this lab are for you to

1. reinforce limit calculation techniques,

2. re-examine the connection between limits at infinity and horizontal asymptotes, and

3. explore the relationships between position and velocity, and average velocity and instantaneous velocity.

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 part 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="_cstyle27" layout="_pstyle17">Calculating Limits</Text-field>

<Text-field style="_cstyle28" layout="_pstyle18">Limit Laws -- "plugging in" (Direct Substitution)</Text-field>

In Section 2.3, Calculating Limits Using the Limit Laws, we concluded that you can calculate some limits by plugging a into the function. For example, since 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 is defined at x = 2, we can calculate the limit as x approaches 2 by plugging in 2 for x everywhere:

However, this doesn't always work. Give an example of a limit that you can NOT evaluate by plugging in.

For example, you can not "plug in" if doing so will cause division by zero.

For each of the limits below, see what happens if you try to "plug in", then examine Maple's evaluation of the limit. Discuss each example with your partner.

fn:=(2^x-1)/x;

subs(x=0,fn);

limit(fn,x=0);

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

Blue partner: lead the discussion of what happened in these last three commands. What are the ramifications for "plugging in"?

fn:=abs(x)/x; subs(x=0,fn);

limit(fn,x=0);

Purple partner: lead a discussion of what happened in these last three commands. What are the ramifications for "plugging in"?

fn:=(1+x)^(1/x); subs(x=0,fn); limit(fn,x=0);

fn:=(x^3-1)/(x^2-1); subs(x=1,fn); limit(fn,x=1);

Blue partner: what happened here?

fn:=((1+h)^4-1)/h; subs(h=0,fn); limit(fn,h=0);

Could you plug in h = 0? Does the limit exist as h approaches 0?

fn:=((2+h)^3-8)/h; subs(h=0,fn); limit(fn,h=0);

T/F: When a is not in the domain of f(x), it is impossible to calculate 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. Why/ why not?

In each case the limit of the function as x approaches the desired number ("a") exists, even though a is not in the domain of the function.

Keep working through this lab to learn/ review how to evaluate some of these types of limits....

<Text-field style="_cstyle28" layout="_pstyle18">Eliminating 0 from the denominator</Text-field>

When "plugging in" to evaluate a limit results in division by zero, you can often evaluate the limit by expanding or factoring and cancelling terms. Let's re-examine the last three limits in the previous section. We'll go through the steps you would take to evaluate this limit by hand, using Maple to save a little time when expanding or factoring.

fn:=(x^3-1)/(x^2-1);

factor(x^3-1); factor the numerator

factor(x^2-1); factor the denominator

Observe which term is in both the numerator and in the denominator, and cancel it. (This is the key in evaluating this limit by hand.) So we get

simplify(fn);

Division by zero is no longer a problem if we plug in x = 0, so let's do it: (Recall % gives you your last output. Be sure that you last entered the line above!)

subs(x=0,%);

This agrees with the previously-Maple-calculated limit of 1.

fn:=((1+h)^4-1)/h;

expand((1+h)^4-1); expand (multiply out) the numerator

factor(%); factor the expanded numerator

Do any terms cancel this time?

%/h;

limit(%,h=0);

fn:=((2+h)^3-8)/h;

expand((2+h)^3-8);

factor(%);

From this point you can mentally cancel the h in the denominator with the h showing in the expanded and factor version of the numerator. What's left is a polynomial in h: plug in h = 0 to see that the limit is 12.

<Text-field style="_cstyle28" layout="_pstyle18">Left-hand and right-hand limits via Maple</Text-field>

Occasionally you may want to use Maple to calculate a right-hand or left-hand limit. The command is virtually the same as the limit command, you just add right or left.

limit(x^x,x=0,right);

Examine the graph to predict the left-hand and right-hand limits as x approaches 0.

f:=x->abs(x)/x;

plot(f(x),x=-3..3,y=-2..2,discont=true);

From the graph, what do you expect 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 to be?

Type in the commands to calculate the left and the right hand limits as x approaches 0.

Do you hold the same expectation for limit(abs(x), x = 0)? Enter the command to verify that you are right.

<Text-field style="ParagraphStyle10" layout="_pstyle17"><Font style="_cstyle27">Limits at Infinity</Font></Text-field>

<Text-field style="Heading 2" layout="Heading 2">The connection between limits at infinity and horizontal asymptotes</Text-field>

Recall, from Section 2.6, Definition 2:

The line y = L is a horizontal asympotote of the curve y = f(x) if either 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 or LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2OVEhRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lJXNpemVHUSMxNEYnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHUSV0cnVlRicvJSp1bmRlcmxpbmVHRjcvJSpzdWJzY3JpcHRHRjcvJSxzdXBlcnNjcmlwdEdGNy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStiYWNrZ3JvdW5kR0ZDLyUnb3BhcXVlR0Y3LyUrZXhlY3V0YWJsZUdGNy8lKXJlYWRvbmx5R0Y3LyUpY29tcG9zZWRHRjcvJSpjb252ZXJ0ZWRHRjcvJStpbXNlbGVjdGVkR0Y3LyUscGxhY2Vob2xkZXJHRjcvJTBmb250X3N0eWxlX25hbWVHUSgyRH5NYXRoRicvJSptYXRoY29sb3JHRkMvJS9tYXRoYmFja2dyb3VuZEdGQy8lK2ZvbnRmYW1pbHlHRjEvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUpbWF0aHNpemVHRjQtRiM2KUYrLUknbXVuZGVyR0YkNiUtSSNtb0dGJDYzUSRsaW1GJy8lJWZvcm1HUSdwcmVmaXhGJy8lJmZlbmNlR0Y3LyUqc2VwYXJhdG9yR0Y3LyUnbHNwYWNlR1EkMGVtRicvJSdyc3BhY2VHUS50aGlubWF0aHNwYWNlRicvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGNy8lKG1heHNpemVHUSlpbmZpbml0eUYnLyUobWluc2l6ZUdRIjFGJy8lKGxhcmdlb3BHRjcvJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y3LyUwZm9udF9zdHlsZV9uYW1lR0ZWLyUlc2l6ZUdGNC8lK2ZvcmVncm91bmRHRkMvJStiYWNrZ3JvdW5kR0ZDLUYjNiYtRiw2OVEieEYnRi9GMkY1RjhGO0Y9Rj9GQUZERkZGSEZKRkxGTkZQRlJGVEZXRllGZW5GZ25Gam4tRmJvNjNRJyZyYXJyO0YnL0Zmb1EmaW5maXhGJ0Zob0Zqby9GXXBRL3RoaWNrbWF0aHNwYWNlRicvRmBwRmVyRmJwRmRwRmZwRmlwRlxxL0ZfcUY3RmBxRmJxRmRxRmZxRmhxLUYjNiQtRmJvNjNRKiZ1bWludXMwO0YnRmVvRmhvRmpvRlxwL0ZgcEZecEZicEZkcEZmcEZpcEZccUZnckZgcUZicUZkcUZmcUZocS1GLDY5RmhwRi9GMkY1RjhGO0Y9Rj9GQUZERkZGSEZKRkxGTkZQRlJGVEZXRllGZW5GZ25Gam5GKy8lLGFjY2VudHVuZGVyR0Y3LUYsNjlGLkYvRjJGNUY4RjtGPUY/RkEvRkVRLlsyNTUsMjU1LDI1NV1GJ0ZGRkhGSkZMRk5GUEZSRlRGVy9GWkZlc0ZlbkZnbkZqbi1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EnMC4wfmV4RicvJSZ3aWR0aEdRJzAuNH5lbUYnLyUmZGVwdGhHRlx0LyUqbGluZWJyZWFrR1ElYXV0b0YnRistRiM2Ji1GLDY5USJmRidGL0YyRjVGOEY7Rj1GP0ZBRkRGRkZIRkpGTEZORlBGUkZURldGWUZlbkZnbkZqbi1GYm82M1EwJkFwcGx5RnVuY3Rpb247RidGYnJGaG9Gam9GXHBGXXNGYnBGZHBGZnBGaXBGXHFGZ3JGYHFGYnFGZHFGZnFGaHEtRiM2JS1GYm82M1EiKEYnRmVvL0Zpb0Y6RmpvL0ZdcEZhcEZfcC9GY3BGOkZkcEZmcEZpcEZccUZnckZgcUZicUZkcUZmcUZocS1GIzYjRlxyLUZibzYzUSIpRicvRmZvUShwb3N0Zml4RidGYnVGam9GY3UvRmBwUTJ2ZXJ5dGhpbm1hdGhzcGFjZUYnRmR1RmRwRmZwRmlwRlxxRmdyRmBxRmJxRmRxRmZxRmhxRitGKy1GYm82M1EiPUYnRmJyRmhvRmpvRmRyRmZyRmJwRmRwRmZwRmlwRlxxRmdyRmBxRmJxRmRxRmZxRmhxLUYsNjlRIkxGJ0YvRjJGNUY4RjtGPUY/RkFGREZGRkhGSkZMRk5GUEZSRlRGV0ZZRmVuRmduRmpu.

If you're comfortable with this definition, feel free to skip the next subsection and go on to the problems. (Take a look at any comments received on your work on Lab 2, #4 as well.)

<Text-field style="Heading 2" layout="Heading 2">explanatory section repeated from Lab 2 (with HW advice)</Text-field>

If you had trouble with #4 on Lab 2, please reread (and process) the text. When you get to the homework, be sure to read the directions! Provide graph(s), the limit calculations, and words to back up your conclusions....

Remember learning about horizontal asymptotes in Precalculus? 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);

WHAT DOES THIS LIMIT CALCULATION TELL YOU ABOUT THE EXISTENCE OF A HORIZONTAL ASYMPOTE?

plot({(x+1)/(x-1),1},x=100..1000);

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

<Text-field style="_cstyle39" layout="_pstyle17">Lab 3 Homework</Text-field>

Submission Instructions: Obtain the worksheet "Lab 3 Homework" from BlackBoard. You will edit this worksheet, then print it and turn in one copy per lab team when due. (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.

1. A group of CSUCI friends decide to spend a weekend in Buellton, CA. They jump in a car and take off. As they're going through Oxnard, they realize they forgot one friend! Embarrassed, they drive slowly back to CSUCI and get their friend. All this takes two hours. They then proceed to Buellton, arriving in another two hours. Their distance (in miles) from CSUCI at time t (in hours) is given by 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 .

a) Enter the position function p(t) into Maple and graph it for the times between t = 0 when the friends first left CSUCI and t = 4 when they arrived in Buellton. You'll paste this graph into your worksheet later.

b) Approximately when did the friends arrive in Oxnard? According to the information given, what is the distance from CSUCI to Oxnard?

c) What is the exact distance (according to the information given) between CSUCI and Buellton?

d) Find the friends' average velocity from

i) CSUCI to Oxnard (the first time)

ii) Oxnard back to CSUCI

e) Find the friends' average velocity for the following time periods:

i) 3 to 3.5 hours

ii) 3 to 3.1 hours

iii) 3 to 3.01 hours

iv) 3 to 3.001 hours

f) Find the friends' average velocity for the following time periods:

i) 0.5 to 1 hour

ii) .9 to 1 hour

iii) .99 to 1 hour

iv) .999 to 1 hour

g) What do you think the friends' instantaneous velocity was (what did their speedometer read) at

i) t = 3 hours?

ii) t = 1 hour?

h) Copy and paste your graph from part a) into the Lab 3 worksheet. When you've printed the worksheet, write in and label the following items:

i) when the friends arrived in Oxnard the first time

ii) when the friends arrived back at CSUCI to get the missing friend

iii) the line with slope corresponding to the friends' average velocity between 3 and 3.5 hours

iv) the line with slope corresponding to the friends' instantaneous velocity 3 hours into their trip

v) the line with slope corresponding to the friends' average velocity between 0.5 and 1 hour

vi) the line with slope corresponding to the friends' instantaneous velocity 1 hour into their trip

2. Use Maple -- if you wish -- to help with the algebra. Evaluate 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 by following the steps outlined.

a) expand the numerator

b) factor the expanded numerator

c) divide the factored, expanded numerator by h

d) plug in h = 0.

e) Answer: 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 = ??

3. Use Maple -- if you wish -- to help with the algebra. Evaluate 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 as outlined.

a) factor the numerator

b) factor the denominator

c) recombine the numerator and denominator, and cancel any common factors

d) plug in x = 3

e) Answer: 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 = ??

4. Exploration problem: Consider the function 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 and the limit 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.

a) Is this function defined at 0?

b) We know LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2OVEhRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lJXNpemVHUSMxMkYnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHUSV0cnVlRicvJSp1bmRlcmxpbmVHRjcvJSpzdWJzY3JpcHRHRjcvJSxzdXBlcnNjcmlwdEdGNy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStiYWNrZ3JvdW5kR0ZDLyUnb3BhcXVlR0Y3LyUrZXhlY3V0YWJsZUdGNy8lKXJlYWRvbmx5R0Y3LyUpY29tcG9zZWRHRjcvJSpjb252ZXJ0ZWRHRjcvJStpbXNlbGVjdGVkR0Y3LyUscGxhY2Vob2xkZXJHRjcvJTBmb250X3N0eWxlX25hbWVHUSgyRH5NYXRoRicvJSptYXRoY29sb3JHRkMvJS9tYXRoYmFja2dyb3VuZEdGQy8lK2ZvbnRmYW1pbHlHRjEvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUpbWF0aHNpemVHRjQtRiM2KkYrLUknbXVuZGVyR0YkNiUtSSNtb0dGJDYzUSRsaW1GJy8lJWZvcm1HUSdwcmVmaXhGJy8lJmZlbmNlR0Y3LyUqc2VwYXJhdG9yR0Y3LyUnbHNwYWNlR1EkMGVtRicvJSdyc3BhY2VHUS50aGlubWF0aHNwYWNlRicvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGNy8lKG1heHNpemVHUSlpbmZpbml0eUYnLyUobWluc2l6ZUdRIjFGJy8lKGxhcmdlb3BHRjcvJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y3LyUwZm9udF9zdHlsZV9uYW1lR0ZWLyUlc2l6ZUdGNC8lK2ZvcmVncm91bmRHRkMvJStiYWNrZ3JvdW5kR0ZDLUYjNiYtRiw2OVEieEYnRi9GMkY1RjhGO0Y9Rj9GQUZERkZGSEZKRkxGTkZQRlJGVEZXRllGZW5GZ25Gam4tRmJvNjNRJyZyYXJyO0YnL0Zmb1EmaW5maXhGJ0Zob0Zqby9GXXBRL3RoaWNrbWF0aHNwYWNlRicvRmBwRmVyRmJwRmRwRmZwRmlwRlxxL0ZfcUY3RmBxRmJxRmRxRmZxRmhxLUklbXN1cEdGJDYlLUkjbW5HRiQ2OVEiMEYnRi9GMkY1L0Y5RjdGO0Y9Rj9GQUZERkZGSEZKRkxGTkZQRlJGVEZXRllGZW4vRmhuUSdub3JtYWxGJ0Zqbi1GYm82M1EnJnBsdXM7RidGYnJGaG9Gam8vRl1wUTBtZWRpdW1tYXRoc3BhY2VGJy9GYHBGZnNGYnBGZHBGZnBGaXBGXHFGZ3JGYHFGYnFGZHFGZnFGaHEvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRisvJSxhY2NlbnR1bmRlckdGNy1GLDY5Ri5GL0YyRjVGX3NGO0Y9Rj9GQUZERkZGSEZKRkxGTkZQRlJGVEZXRllGZW5GYHNGam4tSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJzAuMH5leEYnLyUmd2lkdGhHUScwLjR+ZW1GJy8lJmRlcHRoR0ZkdC8lKmxpbmVicmVha0dRJWF1dG9GJ0YrLUkmbWZyYWNHRiQ2Ki1GXHM2OUZbcUYvRjJGNUZfc0Y7Rj1GP0ZBRkRGRkZIRkpGTEZORlBGUkZURldGWUZlbkZgc0ZqbkZcci8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGZ3UvJSliZXZlbGxlZEdGN0ZmcUZocS1GYm82M1EiPUYnRmJyRmhvRmpvRmRyRmZyRmJwRmRwRmZwRmlwRlxxRmdyRmBxRmJxRmRxRmZxRmhxLUZcczY5USgmaW5maW47RidGL0YyRjVGX3NGO0Y9Rj9GQUZERkZGSEZKRkxGTkZQRlJGVEZXRllGZW5GYHNGam5GKw==, and 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 for values of c larger than 1. Do these facts suggest anything about 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?

c) Use Maple to calculate 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, 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, 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and 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. Make a conjecture about 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.

d) Use Maple to calculate LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2OVEhRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lJXNpemVHUSMxMkYnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHUSV0cnVlRicvJSp1bmRlcmxpbmVHRjcvJSpzdWJzY3JpcHRHRjcvJSxzdXBlcnNjcmlwdEdGNy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStiYWNrZ3JvdW5kR0ZDLyUnb3BhcXVlR0Y3LyUrZXhlY3V0YWJsZUdGNy8lKXJlYWRvbmx5R0Y3LyUpY29tcG9zZWRHRjcvJSpjb252ZXJ0ZWRHRjcvJStpbXNlbGVjdGVkR0Y3LyUscGxhY2Vob2xkZXJHRjcvJTBmb250X3N0eWxlX25hbWVHUSgyRH5NYXRoRicvJSptYXRoY29sb3JHRkMvJS9tYXRoYmFja2dyb3VuZEdGQy8lK2ZvbnRmYW1pbHlHRjEvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUpbWF0aHNpemVHRjQtRiM2KUYrLUknbXVuZGVyR0YkNiUtSSNtb0dGJDYzUSRsaW1GJy8lJWZvcm1HUSdwcmVmaXhGJy8lJmZlbmNlR0Y3LyUqc2VwYXJhdG9yR0Y3LyUnbHNwYWNlR1EkMGVtRicvJSdyc3BhY2VHUS50aGlubWF0aHNwYWNlRicvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGNy8lKG1heHNpemVHUSlpbmZpbml0eUYnLyUobWluc2l6ZUdRIjFGJy8lKGxhcmdlb3BHRjcvJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y3LyUwZm9udF9zdHlsZV9uYW1lR0ZWLyUlc2l6ZUdGNC8lK2ZvcmVncm91bmRHRkMvJStiYWNrZ3JvdW5kR0ZDLUYjNiYtRiw2OVEieEYnRi9GMkY1RjhGO0Y9Rj9GQUZERkZGSEZKRkxGTkZQRlJGVEZXRllGZW5GZ25Gam4tRmJvNjNRJyZyYXJyO0YnL0Zmb1EmaW5maXhGJ0Zob0Zqby9GXXBRL3RoaWNrbWF0aHNwYWNlRicvRmBwRmVyRmJwRmRwRmZwRmlwRlxxL0ZfcUY3RmBxRmJxRmRxRmZxRmhxLUklbXN1cEdGJDYlLUkjbW5HRiQ2OVEiMEYnRi9GMkY1L0Y5RjdGO0Y9Rj9GQUZERkZGSEZKRkxGTkZQRlJGVEZXRllGZW4vRmhuUSdub3JtYWxGJ0Zqbi1GYm82M1EnJnBsdXM7RidGYnJGaG9Gam8vRl1wUTBtZWRpdW1tYXRoc3BhY2VGJy9GYHBGZnNGYnBGZHBGZnBGaXBGXHFGZ3JGYHFGYnFGZHFGZnFGaHEvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRisvJSxhY2NlbnR1bmRlckdGNy1GLDY5Ri5GL0YyRjVGX3NGO0Y9Rj9GQUZERkZGSEZKRkxGTkZQRlJGVEZXRllGZW5GYHNGam4tSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJzAuMH5leEYnLyUmd2lkdGhHUScwLjR+ZW1GJy8lJmRlcHRoR0ZkdC8lKmxpbmVicmVha0dRJWF1dG9GJ0YrLUYjNiVGKy1GIzYlLUYsNjlRImZGJ0YvRjJGNUY4RjtGPUY/RkFGREZGRkhGSkZMRk5GUEZSRlRGV0ZZRmVuRmduRmpuLUZibzYzUTAmQXBwbHlGdW5jdGlvbjtGJ0ZickZob0Zqb0ZccC9GYHBGXnBGYnBGZHBGZnBGaXBGXHFGZ3JGYHFGYnFGZHFGZnFGaHEtSShtZmVuY2VkR0YkNiMtRiM2Iy1GLDY5Rl5yRi9GMkY1RjhGO0Y9Rj9GQS9GRVEuWzI1NSwyNTUsMjU1XUYnRkZGSEZKRkxGTkZQRlJGVEZXL0ZaRmB2RmVuRmduRmpuLUZibzYzUTEmSW52aXNpYmxlVGltZXM7RicvRmZvRi5GaG9Gam9GXHBGZ3VGYnBGZHBGZnBGaXBGXHFGZ3JGYHFGYnFGZHFGZnFGaHFGK0Yrand examine the graph of 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 for 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 in 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. Comment on your conjecture and any thoughts connected with part b).

5. Analyze the graph and the limits resulting from entering the commands below. Refer to both graphs and limits, as well as to any other graphs, equations, etc. you find useful in explaining whether the function has any horizontal asymptotes, and how you know. (You're not supposed to be familiar with the function -- don't worry if it looks strange!)

f:=x->piecewise(x<-10 or x = -10, exp(x+10)+2, x > -10 and (x<1 or x = 1),1/10000*exp(-x-1)+2,x>1,cos(1/(x-1))*(Beta(x,2)+3));

plot(f(x),x,discont=true);

(Copy and paste the plot command, and put in different values of x to examine different parts of the graph. For example, to see what happens for x in [10, 20], use

plot(f(x),x=10..20,discont=true);

<Text-field style="_cstyle41" layout="_pstyle17">Acknowledgements</Text-field>

Lab written by C. Wyels and H. King, Sept. '01, updated by C. Wyels, F'06. Function and story for problem #1 adapted from p. 202 of CalcLabs with Maple V, Boggess, Barrow, et al, Brooks/Cole, 1995.