Mathematics 161                                  Laboratory 3                             September 17, 1992
	  
Name: _____________________________   Lab Partner: ___________________________
	  
Consulted with:  ____________________________________________________________

© Lafayette College, 1994

Estimating Limits

Today, we will investigate graphical and numerical techniques for estimating the limits of transcendental functions (harder than polynomials and rationals).

Part 0: Tables of Function Values

In other labs you have used Mathematica’s Plot command to examine the graphs of functions.  The Table command can be used to create a table of numerical values of a function.  For example, if we wanted to know the values of the sine function at the first 11 nonnegative integers, we could use the command

T = Table[{n, Sin[n]}, {n, 0, 10}]

to create the list of ordered pairs (0,0), (1, sin(1)), ... , (10, sin(10)).  We can force the computer to actually evaluate the sine function at each of the values of n by following this command with

N[T]

(N[expr] returns the numerical value of expr.)

Part 1:  A Familiar Function

Define the function f[x_] := Sin[x]/x.  Recall, we investigated this function in Part 5 of last week’s lab.  We will use a table of values of f(x) to investigate the limit of this function as x approaches 0.

(a)	Use the command

T1 = Table[{1/2^n, f[1/2^n]}, {n, 1, 10}]

to create a table of values of f(x) for x = 1/2, 1/4, ... , 1/210.  Use N[T1] to generate the numerical values.

Modify your Table command to create a table of values of f(x) for negative values of x which approach 0.  Write your command below.  Based on the tables you have made, guess the limit as x approaches zero for f(x):

(b)	To refresh your memory of this function’s graph, use

Plot[f[x], {x, -2Pi, 2Pi}]

to draw the graph of f(x) over the interval [–2π,2π].  Does the graph support your guess in (a)?  (Yes or No)

Part 2: The Sandwich Theorem

In this week’s graded homework, you must evaluate the limit of cos(1/x) as x → 0.  Have Mathematica plot the graph of this function on a relevant interval. What does the graph lead you to believe is the limit?  We will investigate the limit of the function s(x) = xcos(1/x) as x → 0.  Define s in Mathematica, and create (at least two) tables of values to estimate the limit at zero.  Based on your tables, what do you think the limit is? Explain why your tables justify your guess.

To increase confidence in your answer, use the command below to compare the graph of s with those of |x| and –|x|.  Abs[x] is Mathematica’s name for the absolute value function.

Plot[{s[x], Abs[x], -Abs[x]}, {x, -1, 1},
     PlotStyle->{Red, Blue, Green}]

What is the relationship between the graphs of s and the other two functions?  Explain in a couple of sentences why the sandwich theorem holds, and what it tells us about the limit of s at zero.

Part 3:  Some Limits to Estimate

Use tables and plots to evaluate at least one of the following limits.  Give brief explanations supporting your answer(s).

2) Your turn! Define a function that has (at least) one point, a, not in its domain.
    Evaluate the limit of your function as the input approaches a, using tables and plots.
    Be sure to give the definition of your function; there is extra credit for especially 
    interesting functions!
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
   

Part 4:  An Unfamiliar Function

The error function, erf(x)—a function which has no “elementary” transformation rule—arises frequently in the study of probability and statistics and in other areas as well; its domain is the entire real line.  Mathematica knows this function as Erf[x].  (To gain some familiarity with the error function, you might want to make a plot of this function over the interval [–1,1].)

Estimate the limit of the function erf(x)/x as x approaches 0.  Your estimate should be accurate to at least three places after the decimal.  Give a brief explanation of why you believe you have achieved the required accuracy.