# Limits: An Introduction

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.

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