Here we see a consequence of a function being continuous.

We can generalize the example above to get the following theorems.
1 : Where is continuous?
for all real numbers at for all real numbers, except impossible to say
2 : True or false: If and are continuous functions on an interval , then is continuous on .
True False
3 : True or false: If and are continuous functions on an interval , then is continuous on .
True False

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 to the inside of continuous functions.

Using the Composition Limit Law, we can compute the last example from the beginning of this section.

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.

Let’s see some examples!

4 : Can this limit be directly computed by limit laws?
yes no
4.1 : Compute:
5 : Can this limit be directly computed by limit laws?
yes no