1. The limit of a function.

Maple can find lots of limits. Here is the syntax to evaluate the limit .

> limit(sin(x)/x, x=0);

Here is the syntax for and :

> limit(2/(x-3),x=3,left); limit(2/(x-3),x=3,right);

Submission:

Graph the function , then change the scale to 'zoom in' on the y-intercept. Use the graph to estimate . Show an appropriate graph and the value of the limit obtained by using the limit command.  Do the same process with the function .

Submission Worksheet:

2. Plotting the limit of a function.

Consider the function . Let us define this function with Maple.

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

Now attempt to evaluate the function at . You should of course know that this is impossible.

> f(0);

Error, (in f) numeric exception: division by zero

Good! We should really be surprised if Maple would give us an answer.  Now lets ask Maple to graph this function from -1 to 1.

> plot(f,-1..1);

That's interesting! Lets plot the function from -.1 to .1

> plot(f,-0.1..0.1);

That's better. We know that the function is not defined at 0, but the graph seems to pass right through the y-axis in a nice continuous line. In fact what we have here is a removable discontinuity . The function appears to be tending towards a specific value as gets close to 0. Lets compute a sequence of values where approaches 0. Compute f(x) for x = .1, .01, .001, .0001, and -.1, -.01 -.001, -.0001.

> f(.1);

> f(.01);

> f(.001);

> f(.0001);

> f(-.1);

> f(-.01);

> f(-.001);

> f(-.0001);

We see that to three decimal places f(.0001) and f(-.0001) agree.  Even though the function is not defined at 0 the function tends towards a limiting value as x approaches 0. In our example the limit is approximately 2.718.

Go through the routine outlined above for the limits for the following: In each case,

(a) show an appropriate graph,

(b) estimate the limit by appropriate function evaluations.

Submission worksheet:

3. The precise definition of a limit.

We say that f(x) approaches the limit L as x approaches a if for every >0 there is a corresponding >0 such that whenever < . We formally write . The purpose of this activity is to reinforce the meaning of this definition by focusing on specific values for .

Example:
Use a graph to find a number so that whenever .  In the precise definition of a limit, we have , , and . What we need to do is to find a delta so that if x is within delta of 2, then will be within 0.5 of 3 . Let us take a wild guess at ( for our purposes 1 is a wild guess) and plot in the vicinity of ( a,L ).

> epsilon:=0.5; delta:=1; plot([3-epsilon, 3+epsilon, sqrt(4*x+1)], x=2-delta..2+delta, axes=boxed, color=[red, red, blue]);

We see that for this choice of , the -coordinate of the point on the graph gets about 0.6 units away from 3. The prescribed tolerance is , so this is not small enough! Let's try another value of .

> epsilon:=0.5; delta:=0.4; plot([3-epsilon, 3+epsilon, sqrt(4*x+1)], x=2-delta..2+delta, axes=boxed, color=[red, red, blue]);

We see here that the y-coordinate of the point on the curve gets only about 0.3 units away from 3, so now we are being a little stingy, a larger value of could have been chosen but not necessary. Let us try to find a value of which is closer to the maximum allowable value.

> epsilon:=0.5; delta:=0.65; plot([3-epsilon, 3+epsilon, sqrt(4*x+1)], x=2-delta..2+delta, axes=boxed, color=[red, red, blue]);

By studying the above graph, we see that if the distance from to 2 is less than about 0.7, then the distance from to 3 will be less than 0.5.

Submission:

By following and adapting the above method, do the following problems

(a) Find a number such that whenever .

(b) For the limit find a value for that corresponds to

(c) For the limit find a value for delta that corresponds to

Submission Worksheet:

4. Using Maple to evaluate the limit.

If a function has a limiting value Maple will of course evaluate that value.  Here is the syntax to evaluate the limit of our function, , at .

> limit((1+x)^(1/x),x=0);

Yes indeed it is e . We can use the evalf command to get its decimal approximation.

> evalf(%);

Submission:

Use the Maple commands to evaluate the limits of these four functions:

(a)

(b)

(c)

(d)

Submission worksheet:

5. Using Maple to estimate limits and compute limits exactly.

For many functions, one can evaluate by simply plugging the value into the function. A function with this nice property is called continuous at .  But there are many important cases where a limit exists that you cannot find in this simple manner. If you have a good graphing utility, like Maple, you can get an estimate of the limit by looking at a graph of the function. Consider the function f defined by

> f:=x->(x^4-2*x^3-7*x^2+20*x-12)/(x-2)^2;

It is clear that f(2) does not exist, because we would get a zero in the denominator. But let us plot the function and guess about the limit.

> plot(f,1..3);

It sure looks like the limit of as is around 5.  Let us let Maple evaluate this limit exactly.

> limit(f(x),x=2);

So we were able to use Maple to evaluate this limit exactly and the value coincides with our estimate. Keep in mind that whenever you evaluate a number by looking at a graph, your answer is only an estimate. You can never tell the exact answer from a picture.

Submission:

For the following functions and by following and adapting the above method,

(a) Plot the function near the point .

(b) From the graph guess the value of the limit of the function as the function approaches .

(c) Evaluate the limit using the Maple command limit .

(d) How close was your guess?

1. ,

2. ,

3. ,

4. 

Submission worksheet: