Limits of composite functions - How to find them?

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In this video we’re talking about how to find the limit of composite functions.

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0:33 // Two different ways to write a composite
1:01 // Two ways to find the limit of a composite: algebraically and graphically
1:28 // How to find the limit of a composite algebraically
4:10 // How to find the limit of a composite when you have the graphs
6:52 // Composites can be given as one function or as two functions (together vs. separately). How to split a composite function apart? How to create a composite function from two functions?
9:33 // Limit laws still apply when you’re finding the limit of a composite function
11:46 // An example with a rational function
15:46 // What to do when the limit doesn’t exist (DNE). What do you do when the one-sided limits are different? What if the inner function of the composite doesn’t have a general limit?
19:22 // What do do when the outer function of the composite doesn’t have a general limit?
20:44 // What to do when the graph doesn’t have a general limit, or the one-sided limits are different?

Remember that composite functions are “functions of functions”, which means that we have one function plugged into another function. As an example, sin(x^2) is a composite function because we’ve plugged the function x^2 into the function sin(x). Think of any function that as an “outer part” and an “inner part” as composite functions.

The most important thing to know about evaluating limits of composite functions is that we can do so algebraically or graphically. Basically that means we can do math to figure out the limit, or we can look at the graphs of the two functions (if we have them) and figure it out that way. In this video we’ll look at examples of both so that you know how to find the limit using math, and also how to find the limit just by looking at the graphs.

In either case, whether you’re looking at the limit algebraically or graphically, you’ll be finding the limit of the inside function, and then evaluating the outside function at the resulting value. So if you’re taking the limit as x goes to 0 of the composite f(g(x)), that means you’ll be finding the limit as x goes to 0 of g(x), and then taking whatever real-number result you got and plugging that number into f(x). Whatever you get will be the answer to the problem.

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