You are about to erase your work on this activity. Are you sure you want to do this?

Updated Version Available

There is an updated version of this activity. If you update to the most recent version of this activity, then your current progress on this activity will be erased. Regardless, your record of completion will remain. How would you like to proceed?

Mathematical Expression Editor

We give basic laws for working with limits.

In this section, we present a handful of rules called the Limit Laws that allow us to
find limits of various combinations of functions.

Limit Laws Suppose that , , and positive integers, and is a constant.

Sum/Difference Law

.

Constant Multiple Law

.

Product Law

.

Quotient Law

, if .

Power Law

.

Fractional Power Law

, if is nonnegative near when is even and is reduced.

True or false: If and are continuous functions on an interval , then is continuous on
.

TrueFalse

This follows from the Sum/Difference Law.

True or false: If and are continuous functions on an interval , then is continuous on
.

TrueFalse

In this case, will not be continuous for where .

Compute the following limit using limit laws:

Well, get out your pencil and write
with me: by the Sum/Difference Law. So now by the Product Law. Finally by
continuity of and , We can check our answer by looking at the graph of :

We can generalize the example above to get the following theorems.

Continuity of Polynomial Functions All polynomial functions, meaning functions of
the form where is a whole number and each is a real number, are continuous for all
real numbers.

Continuity of Rational Functions Let and be polynomials. Then a rational function,
meaning an expression of the form is continuous for all real numbers except where .
That is, rational functions are continuous wherever they are defined.

Let be a real
number such that . Then, since is continuous at , . Therefore, write with me, and
now by the Quotient Law, and by the continuity of polynomials we may
now set Since we have shown that , we have shown that is continuous at
.

Where is continuous?

for all real numbersat for all real numbers, except impossible to say

Back in Theorem theorem:continuity we mentioned a big list of functions that were continuous. We
mention them again in the following statement. We will study some of these functions
in more detail in later sections. For now, we focus only on the fact that they are
continuous.

Continuity of Other Functions Polynomials, the trigonometric functions
and , and exponential functions are continuous everywhere. Rational functions,
logarithms, and the other trigonometric functions are continuous in their
domain.

Compute the limit:

By the limit laws, . .

Putting these together, .

Now, we give basic rules for how limits interact with composition of functions.

Composition Limit Law If is continuous at , then

Because the limit of a continuous function is the same as the function value, we can
now pass limits inside continuous functions.

Continuity of Composite Functions If is continuous at , then is continuous at
.

Compute the following limit using limit laws:

By continuity of ,
assuming , and now since cosine is continuous for all real numbers,

We can confirm our results by checking out the graph of :

Many of the Limit Laws and theorems about continuity in this section might seem
like they should be obvious. You may be wondering why we spent an entire section
on these theorems. The answer is that these theorems will tell you exactly
when it is easy to find the value of a limit, and exactly what to do in those
cases.

The most important thing to learn from this section is whether the limit laws can be
applied for a certain problem, and when we need to do something more interesting.
We will begin discussing those more interesting cases in the next section. For now, we
end this section with a question:

A list of questions

Let’s try this out.

Can this limit be directly computed by limit laws?

yesno

Compute:

Since is a rational function, and the denominator does not equal , we see
that is continuous at . Thus, to find this limit, it suffices to plug into .

Can this limit be directly computed by limit laws?

yesno

is a rational function, but the denominator equals when . None of our current
theorems address the situation when the denominator of a fraction approaches .

Can this limit be directly computed by limit laws?

yesno

If we are trying to use limit laws to compute this limit, we would first have
to use the Product Law to say that We are only allowed to use this law
if both limits exist, so we must check this first. We know from continuity
that However, we also know that oscillates “wildly” as approaches , and
so the limit does not exist. Therefore, we cannot use the Product Law.

Can this limit be directly computed by limit laws?

yesno

Notice that If we are trying to use limit laws to compute this limit, we
would like to use the Quotient Law to say that We are only allowed to
use this law if both limits exist and the denominator is not . We suspect
that the limit in the denominator might equal , so we check this limit.

This means that the denominator is zero and hence we cannot use the Quotient Law.

Can this limit be directly computed by limit laws?

yesno

Compute:

Notice that If we are trying to use Limit Laws to compute this limit, we
would now have to use the Quotient Law to say that We are only allowed to use this
law if both limits exist and the denominator is not . Let’s check the denominator
and numerator separately. First we’ll compute the limit of the denominator:

Therefore, the limit in the denominator exists and does not equal . We can use
the Quotient Law, so we will compute the limit of the numerator: Hence

Can this limit be directly computed by limit laws?

yesno

If we are trying to use limit laws to compute this limit, we would
have to use the Product Law to say that We are only allowed to
use this law if both limits exist. Let’s check each limit separately.

So this limit exists. Now we check the other factor. Notice that If we are
trying to use limit laws to compute this limit, we would now have to use
the Quotient Law to say that We are only allowed to use this law if
both limits exist and the denominator does not equal . The limit in the
numerator definitely exists, so let’s check the limit in the denominator.

Since the denominator is , we cannot apply the Quotient Law.

Can this limit be directly computed by limit laws?

yesno

If we are trying to use limit laws to compute this limit, we would have to use the
Product Law to say that We are only allowed to use this law if both limits exist. We
know , but what about ? We do not know how to find using limit laws because is
not in the domain of .

Can this limit be directly computed by limit laws?

yesno

Compute:

If we are trying to use limit laws to compute this limit, we would have to use
the Quotient Law to say that We are only allowed to use this law if both limits exist
and the denominator does not equal . Let’s check each limit separately, starting with
the denominator

On the other hand the limit in the numerator is

The limits in both the numerator and denominator exist and the limit in the
denominator does not equal , so we can use the Quotient Law. We find:

Can this limit be directly computed by limit laws?

yesno

We do not have any limit laws for functions of the form , so we cannot compute this
limit.