Lab 1: limits of functions

Math 128 –Calculus
25 points

Lab #02

If you have any questions, first try to find the answer using the help menu within Maple. If that doesn’t work, then raise your hand.

You should see a blank page. First, let’s add a heading. Go to insert  text. Your heading for all your labs should be in the following format:

Name
Lab #02

Date:

Make sure you save your worksheet. Please save your file in the form:

YourLastNameHere-Lab02

Section 1:

1. 
   Input the function f(x) = (1 – 2^x)/x.
   
2. 
   Answer the questions 1 - 3 on your worksheet.
   
3. 
   Plot f(x) for x = -2..2, y = -2..2
   
4. 
   Answer questions 4 – 5 on your worksheet.
   
5. 
   Now, we want to have Maple compute f(x), as a decimal, for x values that are getting closer and closer to 0. To get Maple to convert the answers to decimals, you use the command evalf. For example:
   f(1/2);
   To convert this to a decimal, we input the following (remember that % refers to the last output that Maple gave us).
   evalf(%);
   Alternatively, we could have done the entire thing in one step by entering:
   evalf(f(1/2));
   
6. 
   Using Maple, complete the chart (#6) on the worksheet and then answer question 7.
   

Section 2:

1. 
   Define g(x) to be [(x² + x)/(x² – 1)] * |x² – 1|
   (recall that abs( ) means the absolute value of whatever you put in the parenthesis).
   
2. 
   Plot g(x) on the interval x = -2..2 (let Maple choose the y ranges automatically)
   
3. 
   You should notice that something odd appears to be happening on the graph at x=1. You will shortly learn how to verify the fact that g(x) has a discontinuity at x=1. So far in class, we have seen how some functions have a hole in them. Others, like g(x), can have "jumps" in them. Unfortunately, unless told to look for them, Maple will try to connect the "jumps". We can tell Maple to look for them and show the "jumps" in the graph by typing the following plot command.
   plot(g(x), x=-2..2, discont=true);
   
4. 
   Answer Section 2 questions 1 and 2.
   

We can have Maple compute the limits if they exist. In order to do so, type:
limit(f(x),x=c) This computes the limit of the function f(x) as x approaches c.
limit(f(x),x=c,right) Computes the limit of f(x) as x approaches c from the right
limit(f(x),x=c,left) Computes the limit of f(x) as x approaches c from the left

5. 
   Using your answer(s) from question 2 of section 2, find the limits of g(x) as x approaches c from the left and then from the right (for each of your c values). Reminder, you are using g(x) in part 2, not f(x). Write your answers in for question 3 of Section 2 on the worksheet.
   

You should hand in a copy of your Maple printout and your completed worksheet for Lab #02.

Name __________________________

Math 128 Lab 02 Worksheet

Section 1:

1. 
   Is f defined at x = 0?
   
   
   
2. 
   Why or why not?
   
   
   
3. 
   Using Maple, evaluate f at x =0, what does Maple say the result is?
   
   
   
4. 
   Does the limit of f(x) as x approaches 0 exist?
   
   
   
5. 
   If so, guess a value for that limit.
   
   
   
6. 
   
   

f(x)	Decimal value
f(1/2)
f(1/4)
f(1/8)
f(1/16)
f(1/32)
f(1/64)
f(1/128)
f(1/256)
f(1/512)
f(1/1024)
f(1/2048)

7. 
   Based on the table values, estimate the limit of f(x) as x approaches 0 (to 3 decimal places).
   

Section 2:

1. 
   From your graph of g(x), for what value(s) of c does the limit of g(x) as x approaches c not exist?
   
2. 
   For each value of c from the previous question, estimate the limit as x approaches c from the left and the limit as x approaches c from the right:
   
   1. 
      From the left: From the right:
      

3. 
   For each c value in #2, use Maple to compute the limit as x approaches c from the left and from the right.