There are many rules for computing limits. I'll list the most important ones. There are analogous results for left and right-hand limits; just replace " " with " " or " ".

I mentioned the first rule earlier:

- If is a polynomial, then

Remember that a polynomial is a function of the form

where the a's are numbers. The rule says you can compute the limit of a polynomial as x goes to c by plugging c in for x.

* Example.*

- "The limit of a sum is the sum of the limits."

This equation --- and the ones like it that follow --- must be
interpreted in the right way. The statement is true *provided
that* the two limits on the right side are defined.

* Example.* If I tell you that

then

- "The limit of a product is the product of the limits."

Again, the statement is true *provided that* the two limits on
the right side are defined.

Using mathematical induction and the rule for products, you can prove:

- "The limit of a power is the power of the limit."

In this equation, n is a positive integer, and the statement is true provided that is defined.

* Example.* If I tell you that

then

You have to be a little careful with quotients to avoid division by zero.

- "The limit of a quotient is the quotient of the limits" --- provided that the denominator does not become 0 when you plug in.

If the denominator approaches 0, there are two possibilities.

(a) If the numerator approaches a nonzero number, the limit is
*undefined*.

(b) If the numerator approaches 0, you must do additional work to decide whether the limit is defined (and if it's defined, what its value is).

In any case, the statement is only true if and are defined.

* Example.* Using the rules for quotients and
for polynomials,

As the last example shows, you can use the rules for quotients and
for polynomials to compute the limit of a * rational
function*. A * rational function* is a
polynomial divided by a polynomial. For example,

are rational functions.

* Example.*

Reason: The numerator approaches 6 (a nonzero number) while the denominator approaches 0.

On the other hand,

There's nothing wrong with having 0 on the *top* of a
fraction.

* Example.* Compute .

In this case, plugging in gives , an indeterminate form. I need to do more work to determine whether the limit is defined. Factor and cancel:

- "Constants pull out of limits." If k is a number, then

provided that the limit on the right is defined.

* Example.* If you know that

- "The limit of a root is the root of the limit."

provided that n is an odd positive integer, or n is an even positive integer and is positive.

* Example.* Compute .

Since I'm taking an *odd* root, it doesn't matter whether the
function inside the root is approaching a positive or a negative
number.

* Example.* Compute .

As , the quantity inside the square root approaches . Therefore, the limit is undefined.

* Example.* Compute .

As , I have . You might think that, since
, the limit should be 0. In fact, *the limit is
undefined*.

To see why, remember that when x approaches 3, it does so *from
both sides*. But what happens if x is less than 3? Suppose . Then

In other words, x can't approach 3 from the left (through numbers less than 3) because is undefined for . Hence, the limit is undefined.

Later on, I'll show that if x approaches 3 *from the right*,
then the limit is indeed 0.

The next result is often called the * Sandwich
Theorem* (or the * Squeeze Theorem*). It is
different from the other computational rules in that it produces an
answer in an indirect way.

The * Sandwich Theorem* is an intuitively obvious
result about limits. Suppose you have three functions ,
, , and you're trying to compute the limit of
as x approaches a.

Suppose you know that:

1. and .

2. (at least for x's in some interval around a).

Then

The result in reasonable because g is "sandwiched" between f and h.

* Example.*

As , , but oscillates. And at , is undefined. There's no "algebraic" rule which would allow you to compute the limit, but the Sandwich Theorem makes it easy.

always lies between -1 and 1:

Multiply through by :

Now

Hence, by the Sandwich Theorem

The picture below shows the graphs of , , and . Notice that is indeed caught between the two parabolas, which squeeze the wiggly graph as :

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Copyright 2011 by Bruce Ikenaga