In this video I go over the limit of a composite function theorem which states that if you have a function within another function then the limit can be written outside or within the main function. For example the limit of f(g(x)) equals f(limit of g(x)) as x approaches a. In the proof I used the precise definition of a limit make sure to fully understand it by watching the related videos in the links below.

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Limit of a Composite Function Theorem: Proof

Theorem

If f is continuous at b and:

Then:

Or in other words:

Recap on Precise Definition of a Limit:

If for every number ε > 0 there is a number δ > 0 such that:

Proof

Since f is continuous at b, the limit exists thus the precise definition applies: