Calculus was used long before it was established on firm mathematical
foundations. * Limits* provide a precise way of
talking about * convergence* and infinite
processes.

For example, * derivatives* and *
integrals* are defined using limits. You'll also use limits to
study graphs.

Intuitively, * convergence* means that a variable
quantity *approaches* a fixed number. For example, consider . Plug in numbers close to 2:

It seems as though the -values are close to 0.25. If you graph , the picture seems to confirm this:

Observe that is not defined at .
In thinking about the limit of a function as x approaches c,
you don't consider what happens when x *equals* c; you
consider what happens when x *is close to* c.

In this case, when x is close to 2, it appears that is close to 0.25. The mathematical expression is: {\it The limit of as x approaches 2 is 0.25.} In symbols,

In general, to say that

means that can be made arbitrarily close to L for all x's sufficiently close to a.

I'll discuss the definition and some rules for computing limits later. First, I'll show you some computations so you can get a feel for the ideas.

* Example.* Compute .

If you plug 2 into , you get . This is called an * indeterminate
form*. This means that you can't conclude anything from the form
: The limit might be a number, it might be infinite,
or it might be undefined.

When plugging in yields an indeterminate form, you have to do more work before you can come to a conclusion. "More work" often involves algebraic simplification.

In this case, I fact , then cancel 's:

Why am I allowed to cancel the 's? I noted earlier that in
computing I only
consider x's *near* 2, not x *equal to* 2. Since , I have , so cancellation is legal.

I did the last step by plugging into . This time I did not get an indeterminate form, and the rules for limits I'll discuss later tell me that is the answer.

I won't always describe the action in such excruciating detail, but
you should understand *why* the algebraic manipulations are
legitimate. They usually reduce to the idea in the last example.

* Example.* Compute .

If you plug into , you get . This means you have more work to do.

Since

it follows that

* Example.* Compute .

If I plug into , I get . This is not an indeterminate form; it's just a number. In fact,

The final steps in the last two examples are special cases of the following general rule:

- If is a polynomial, then

That is, you can compute the limit of a polynomial by "plugging
the number in". When you can compute by plugging in (to get ), the function f is
* continuous* at . I'll discuss continuity in
more detail later.

* Example.* Compute .

Plugging in gives . I have more work to do. Add the fractions on the top and simplify:

I got the last equality by plugging 2 into and using the rule for
polynomials. Notice a common thread in the last few problems. If
plugging into produces a form,
*something* must be producing the 0's. Often it is a
*common factor*, which can be *cancelled* from the top
and bottom when you've identified it.

* Example.* Compute .

If you plug into , you get . This means you have some work to do.

First,

Therefore,

I didn't get an indeterminate form when I plugged in, so it's reasonable that the last step is valid. This seems to be confirmed by the graph; .

* Example.* Compute .

Plugging in gives . The limit is *undefined*.

The graph shows a vertical asymptote at :

If I plug in values of x near 1, I get a wide range of outputs:

These empirical results seem to confirm that the limit is undefined.

The general rule is:

- If you plug in and get , the limit is undefined.

Copyright 2005 by Bruce Ikenaga