Theorem for limits of composite functions | Limits and contiuity | AP Calculus | Khan Academy

TL;DR

The video explains how to find the limit of composite functions and states the conditions under which the limit can be equivalently expressed.

Transcript

• [Tutor] In this video, we're going to try to understand limits of composite functions, or at least a way of thinking about limits of composite functions and in particular, we're gonna think about the case where we're trying to find the limit as x approaches a, of f of g of x and we're going to see under certain circumstances, this is going to be ... Read More

Key Insights

• 🎮 Limits of composite functions can be evaluated using the theorem explained in the video.
• ⛔ The existence of the limit of g(x) and the continuity of f(x) at that limit are crucial for applying the theorem.
• ⛔ Graphical representation can help determine the behavior of the functions and their limits.
• 🙃 Point discontinuities do not affect the limit as long as the value of the function approaches the same value from both sides.

Questions & Answers

Q: What are the two conditions required for the limit of f(g(x)) to be equal to f(limit of g(x))?

The two conditions are: 1) the limit of g(x) as x approaches 'a' must exist, and 2) f(x) should be continuous at the limit of g(x).

Q: How is the limit of g(x) calculated in the example?

The limit of g(x) is calculated by evaluating the function as x approaches -3 from both the right and left sides. Both approaches give a value of 3, indicating the existence of the limit.

Q: Is f continuous at the limit of g(x) in the example?

Yes, at x=3, f(x) is continuous according to the graph. Therefore, f meets the condition of being continuous at the limit of g(x).

Q: What is the result obtained by applying the theorem in the example?

By applying the theorem, the limit of f(g(x)) as x approaches -3 is found to be 3, which is also the value of f(g(x)) at the limit.

Summary & Key Takeaways

• The video discusses finding the limit as x approaches 'a' of f(g(x)) for composite functions.

• To use the theorem, the limit of g(x) as x approaches 'a' must exist, and f(x) should be continuous at that limit.

• An example is shown where the limit of f(g(x)) as x approaches -3 is calculated to be 3.