Mathematics 162                                Laboratory 6                          Week of March 1, 1993
	  
Name: _____________________________   Lab Partner: ___________________________
	  
Consulted with:  ____________________________________________________________

© Lafayette College, 1994.

Limits and Indeterminate Forms

In this laboratory you will use numerical methods to examine limits involving indeterminate forms.

Part 0:  A Demonstration

Consider the function

and the problem of computing its limit as x goes to 0.  If x is very close to 0, then both the numerator and denominator of f (x) are close to 0 and it is not clear whether or not the values of f (x) are close to any one particular number.  This limit involves the indeterminate form 0/0.

We can use a graph of f to get a good idea of what the limit is.  From

f[x_] := (1 - Cos[x])/x^2

Plot[f[x], {x, -1, 1}]

we can see pretty clearly that the limit is close to 1/2.  We can try to get more evidence to support our guess by examining a table of values of f.  The command

Table[{t = 0.1^n, f[t]}, {n, 1, 10}]

prints a table of approximate values of f (0.1), f (0.01), f (0.001), ..., f (0.0000000001).  The first few of these values seem to confirm our guess that the limit is 1/2, but something strange happens when we compute f (0.0000001), f (0.00000001), and the others.  Round-off error has crept in and given us very poor accuracy in our computations.  We can overcome this by asking Mathematica to do its arithmetic with more precision.  The following command asks for the same table of function values, but tells Mathematica to use 50 digits in doing its computations.

Table[{t = 1/10^n, N[f[t],50]}, {n, 1, 10}]

This table seems to confirm our guess that

In the following parts of this lab, you should try to get your answers using the first type of table.  But beware of strange output and consider using higher-precision arithmetic if it is needed.

Part 1:  Limits as x Approaches 0

In each of the following, write the indeterminate form which arises in the problem and estimate the indicated limit.  Your estimates should be accurate to three places after the decimal.

The indeterminate form in this limit is:

The indeterminate form in this limit is:

The indeterminate form in this limit is:

The indeterminate form in this limit is:

The indeterminate form in this limit is:

The indeterminate form in this limit is:

Part 2:  Limits as x Approaches Infinity

Once again, you should estimate the value of each limit and indicate the indeterminate form which is involved.  Since we are now interested in values of f (x) for large values of x, we need to modify the table command a bit.  For example, in part a), use the command:

Table[{x = 10.0^n, x^50/E^x}, {n, 1, 10}]

Be sure to notice that x is given as a decimal number (10.0).  This indicates to Mathematica that numerical answers (rather than exact answers, in terms of e) are called for.

The indeterminate form in this limit is:

The indeterminate form in this limit is:

The indeterminate form in this limit is:

The indeterminate form in this limit is:

The indeterminate form in this limit is:

Part 3:  Limits of Integrals

Recall that if f  is continuous and positive over the interval [a, b], then the area bounded by y = f (x), y = 0, x = a, and x = b  is

Mathematica will attempt to evaluate this integral (and find the described area) when we enter the command Integrate[f[x], {x, a, b}].

(a) Define the function f (x) = 1/lnx, and plot it over the interval [2, 8].  Set the range of the plot using the option PlotRange->{0,2}.  Print out a copy of the graph, and shade the region bounded by y = f (x), y = 0, x = 2, and x = 5.  Use Mathematica to find the area of this region, and write it below.

(b) Create a table of values for the integral below for t = 10.0n, for n = 1 to 10.

Estimate the limit:

(c) Repeat this technique to estimate the limit: