Connecting limits and graphical behavior | Limits and continuity | AP Calculus AB | Khan Academy

TL;DR

Limits describe the behavior of a function as it approaches a point, but they don't tell us what's happening at that point or in the rest of the function.

Transcript

• [Instructor] So we have the graph of y is equal to g of x right over here. And I wanna think about what is the limit as x approaches five of g of x? Well we've done this multiple times. Let's think about what g of x approaches as x approaches five from the left. g of x is approaching negative six. As x approaches five from the right, g of x looks... Read More

Key Insights

• ⛔ Limits describe the behavior of a function as it approaches a specific value, but not what happens at that value or elsewhere in the function.
• ⛔ Different functions can have the same limit at a point, highlighting the importance of understanding limits in analyzing function behavior.
• 😥 Limits can be calculated at any interesting point in a function, not just at points with special characteristics.

Questions & Answers

Q: What does a limit describe in a function?

A limit describes the behavior of a function as it approaches a point, but it doesn't tell us what happens at that point or in the rest of the function.

Q: Can different functions have the same limit at a point?

Yes, different functions can have the same limit at a point, even if their graphs look very different.

Q: Can you give an example of two functions with the same limit at a point?

Yes, for example, a function that approaches negative six as x approaches five from both the left and the right, and another function that approaches negative six as well, but with a completely different shape.

Q: Can limits be calculated at any point in a function?

Yes, you can calculate limits at any point in a function, even if it's not a point of discontinuity or any other special characteristic.

Summary & Key Takeaways

• The limit of a function as x approaches five from both the left and the right is approximately negative six.

• Different functions can have the same limit at a point, even if they look very different.

• When finding limits, you can choose any interesting value for x, and the limit will reflect the behavior of the function as x approaches that value.