1. The Limit of a function and graphs.

Graphing alone cannot resolve all limit problems. Consider the function .

This function is of course not defined at 0, but it may have a limit there. We construct graphs of the function with sequentially narrower viewing rectangles (i.e. we zoom in on the origin).

> f:=t->(sqrt(t^2+9)-3)/t^2;

> plot(f(t),t=-5..5,axes=boxed);

> plot(f(t),t=-0.1..0.1,axes=boxed);

> plot(f(t),t=-10^(-6)..10^(-6),axes=boxed);

> plot(f(t),t=-10^(-7)..10^(-7),axes=boxed);

The source of the problem here is the lack of numerical precision. As a default, Maple uses 10 decimal places of precision in its computations.

> Digits;

> f(.0001);f(.00001);

We know that f(.00001) is really not equal to zero. Let's try to fix this by increasing the precision.

> Digits:=25;

> f(.0001);f(.00001);

Will the graphs improve?

> plot(f(t),t=-10^(-6)..10^(-6),axes=boxed);

> plot(f(t),t=-10^(-8)..10^(-8),axes=boxed);

> plot(f(t),t=-10^(-11)..10^(-11),axes=boxed);

You see we have improved things to some extent, but we must expect that the graph problem will emerge again when we zoom in on the origin.

Submission:

(a) Evaluate for and = 1, 0.5, 0.1, 0.05, 0.01, and 0.005 .

(b) Make an educated guess at the value of .

(c) Evaluate h(x) for successively smaller values of x until you finally reach 0 values for h(x). Are you still confident that your guess in part (b) is correct?

(d) Graph h(x) with x=-1..1 . Then zoom in on the origin, until distortions appear.

Submission worksheet: