Functions continuous at specific x-values | Limits and continuity | AP Calculus AB | Khan Academy

TL;DR

The continuity of functions at x=3 is determined by evaluating the limit and value of the function at that point.

Transcript

• [Voiceover] Which of the following functions are continuous at x equals three? Well, as we said in the previous video, in the previous example, in order to be continuous at a point, you at least have to be defined at that point. We saw our definition of continuity, f is continuous at a, if and only if, the limit of f as x approaches a is equal to... Read More

Key Insights

• 😥 Continuity at a point requires the function to be defined and have the limit equal to the function value at that point.
• ❎ The natural logarithm function is not defined for zero or negative numbers.
• #️⃣ The function e^(x-3) is continuous for all real numbers, including x=3.

Questions & Answers

Q: What is the definition of continuity for a function at a specific point?

A function is continuous at a point if and only if the limit of the function as x approaches that point is equal to the value of the function at that point.

Q: Why is the function ln(x-3) not continuous at x=3?

The function ln(x-3) is not continuous at x=3 because it is not defined at that point. The natural logarithm function is not defined for negative numbers, and ln(0) is undefined.

Q: How can we determine the continuity of the function e^(x-3) at x=3?

The function e^(x-3) is continuous at x=3 because it is defined for all real numbers. Additionally, the limit of e^(x-3) as x approaches 3 is e^(3-3) which simplifies to e^0, or 1.

Q: Can we visually understand the discontinuity of ln(x-3) and the continuity of e^(x-3) at x=3?

Yes, if we graph the functions, we can observe the behavior. The graph of ln(x-3) has a gap at x=3 and is not even defined to the left of 3. On the other hand, the graph of e^(x-3) is a smooth, continuous curve with no gaps or jumps.

Summary & Key Takeaways

• In order for a function to be continuous at a point, it must be defined at that point.

• A function is continuous at x=3 only if the limit as x approaches 3 of the function is equal to the value of the function at x=3.

• The natural log function shifted over by 3 (ln(x-3)) is not continuous at x=3 because it is not defined at that point.

• The function e^(x-3) is continuous at x=3 because it is defined for all real numbers and its limit as x approaches 3 evaluates to 1.