We give basic laws for working with limits.

using continuity of the functions , , and , at . Does this imply that we can compute the limits

Well, we cannot use continuity here, because we don’t know if the functions , , , and are continuous at , and

we have no other tools available, since the graphs and tables are not reliable. Obviously, we need more tools to help us with computation of 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.

In plain language, “limit of a sum equals the sum of the limits,” “limit of a product equals the product of the limits,” etc.

Let’s examine how the Limit Laws can be used in computation of limits.

For , write:

For , using the Product Law, we can write:

For , write: The function is a combination of the functions and , but it is neither a sum/difference, nor a product, nor a quotient of these two functions, so we cannot apply any of the Limit Laws. This function is a composition of the two functions, and .

- (a)
- Compute the following limit using limit laws:
- (b)
- Is the polynomial function continuous at ? We have to check whether Since we already know that , we only have to compute .
Therefore, the polynomial function is continuous at .

But what about continuity at any other value ? Is the function continuous on its entire domain?

And what about any other polynomial function?

Are polynomials continuous on their domains?

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

**All polynomial functions**, meaning functions of the form where is a positive integer and each coefficient , , is a real number, are

**continuous for all real numbers**.

**rational function**h, meaning a function of the form where and are polynomials, is

**continuous**for all real numbers except where . That is, rational functions are continuous wherever they are defined.

This means that and .

Now, define a new function, , where , for all in . We have to show that is continuous at , or that

Lat’s start with

and, therefore,

We have proved that is continuous at any number in . Therefore, is continuous on . Similarly, we can prove that is continuous on any interval , by showing it is left-or right-continuous at the endpoints. We can adjust the proof for the function .

We still don’t know how to compute a limit of a composition of two functions. Our next theorem provides basic rules for how limits interact with composition of functions.

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

If is continuous at , and if is continuous at , then is continuous at .

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.

### A list of questions

Let’s try this out.

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

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

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.

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: