The Limits of Composite Functions

So, you obviously know by now that a composite function, or a composition of functions, is just a function with another function inside.

Consider any following function, for now we'll just take a general form of

This means that f is a function that depends on g(x) as the input.

What if we want to find a limit for this ?

Obviously, f is not a function that somehow depends on x, g is.

So, is the same as

However,

Even if the limit of a part of the composition may exist, the limit of the entire composition may not exist!

Let's explain this concept in an easy to understand way. Let's start with these functions:Let's say that you are tasked with finding the limit of as x approaches :This is the same as finding

It is important to note that this limit does not exist in the real numbers, while we could express the square root of -4 as a complex number.

Great job so far. It's time to take a little break again. Just, a deep breath, stand up and move around for a few minutes, and relax your eyes.

When you're done, you can move on to Landau Notation and Local Comparison of Functions - The little o.