MATH 133 -- Intensive Calculus for Science 1

Limit Examples -- October 2, 2001

Consider the function . Note that f(0) is

not defined. However, the following plot seems to indicate

that f( x ) approaches 1/2 as x approaches 0 here:

> f:=x->(exp(x) - x - 1)/x^2:

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

Let's look at this in more detail. To say that the limit of f( x ) as x-> 0

is 1/2, we have to see that for every y-range: < y < there is

a corresponding x-range: < 0 + with the property that

the graph for x in the x-range leaves the plotting window defined

by the y-range through the sides, and not through the top or the bottom.

For instance, let's try , so the y-range is .4 < y < .6

> plot(f(x),x=-1..1,y=0.4..0.6,axes=boxed);

The graph is leaving through the top and bottom, which says that has to be

smaller than 1. If we cut the x-range down to -.4 < x < .4, though, the graph

leaves through the sides:

> plot(f(x),x=-0.4..0.4,y=0.4..0.6,axes=boxed);

BUT, this is only one y-range. To say that the limit is 1/2, we need to

be able to find a corresponding x-range for every y-range (no matter

how small). Let's try a much smaller one:

> plot(f(x),x=-0.01..0.01,y=0.4999..0.5001,axes=boxed);

Note, we have already cut x-range down a lot from the previous graph, but it

still too big to have the graph leave the graphing window through the sides.

We need much smaller:

> plot(f(x),x=-0.0005..0.0005,y=0.4999..0.5001,axes=boxed);

At this point (not before!) we are finally seeing that Maple has tried to evaluate

at x = 0 to draw the graph. That's the reason for the vertical "squiggle" in

the graph. From the mathematical point of view, we really should be looking

at x-ranges with c =0 removed.