1. Limits at Infinity.

The precise definition of a limit at is if and only if for every >0 there is a corresponding such that whenever .
Consider . Let us illustrate the definition for by finding a value of such that if then . To do this, let us plot the function

> f:=x->sqrt(4*x^2+1)/(x-1);

over a domain that will reveal the appropriate value of . Let us first plot the function on the domain 10..200.

> epsilon:=0.05;N:=10;plot([2-epsilon,2+epsilon,f(x)],x=N..200,color=[red,red,blue]);

What this plot suggests is that the value of the function gets outside of the region we want when x gets to be a little more than 40. Therefore, we should choose N to be about 42. Let us illustrate this fact with a better plot.

> epsilon:=0.05;N:=42;plot([2-epsilon,2+epsilon,f(x)],x=N..200,color=[red,red,blue],axes=boxed);

It would appear that when we can take N to be 42.

A good question is how do we really know that as x gets very large (in other words, change the number to a larger number), then the y value will stay within the range 2.05..1.95 ? The answer is that we can not be sure from the plot alone. It is typical that a plot will suggest what the correct answer is, but not allow us to be sure that the answer is correct. One method of verifying the results of the plot is to ask Maple to solve the inequality , which it can do for us.

> solve(abs(sqrt(4*x^2+1)/(x-1)-2) < .05,x);

Maple says that in the open interval (41.12158615, ), this inequality will hold. Thus the value will satisfy the requirements of the definition.

Submission:

Using the methods outlined above find values for that correspond to and for the limit

( Include a graph like the one above for each of the two values of . Also notice that in the definition we must replace with , and we should expect to be negative.)

Submission worksheet:

2. More limits at Infinity.

The formal definition of a limit at is not given in th e text, but can be stated as follows.

Suppose that a function f is defined on an open interval including (that is an interval of the form ( )). Then we say that the limit of as is L , and write , if for every number >0 there is a corresponding positive number , such that whenever .  If , then the line is a horizontal asymptote of the curve . There are several ways to explore limits at infinity using Maple. Let us consider the following function.

> f:=x->1+sin(x)/x;

First, Maple can find limits at infinity by using the limit command.

> limit(f(x),x=infinity);

One can even make a plot which gives an idea of the limit at infinity.

> plot(f(x),x=-infinity..infinity,y=0..2);

One disadvantage of this form of display is that it does not show the value of the limit. Another way to get a display of the limit at infinity is to compose the function with the tangent function and note that as , . So if you plot , you get a graph that shows the limit as . Also when x->-Pi/2, tan(x)->-infinity, so you can determine from the picture as well

> plot(f(tan(x)),x=-Pi/2..Pi/2);

We will use this idea of composing with the tangent function to test the formal definition of the limit at infinity. First, let us use the picture above to guess that . Then let us choose , and plot the lines and on the same graph.

> L:=1;epsilon:=.1;

> plot([f(tan(x)),L-epsilon,L+epsilon],x=0..Pi/2,y=0..2,color=[blue,red,red]);

Now, clicking on the graph, it appears that when x>1.46, the value of the function always lies between these two lines. But since we are plotting the composition of the function and the tangent, this means that when x>tan(1.46), the value of the function lies in the required region. Let us approximate using maple.

> tan(1.46);

We see that if we choose in the formal definition, then whenever .

Finally, let us graph the function over a finite interval, along with its asymptote . One can see that the function appears to get very close to the asymptote, even though it oscillates above and below it.

> plot([f(x),1], x=0..100);

Submission:

Use the ideas above to study the behavior as and for the following functions:

(a)

(b)

(c)

For each function, include a Maple computation of the limit, a graph which allows you to find an N such that whenever , as well as your computation of an N which satisfies the requirement, and a graph which shows the function and its horizontal asymptote as in the same plot.

Submission worksheet: