2011 Calculus AB free response #6a | AP Calculus AB | Khan Academy

TL;DR

The function f(x) is shown to be continuous at x=0 using the limit definition of continuity.

Transcript

Problem number six. Let f be defined by f of x is equal to-- and we have two cases. When x is less than or equal to 0, f is 1 minus 2 sine of x. When x is greater than 0, f is e to the negative 4x. Show that f is continuous at x equals 0. So for something to be continuous at x equals 0, let's think about what has to happen. So if I have a function ... Read More

Key Insights

• 👈 Continuity of a function at a particular point requires the function value, the limit from the left, and the limit from the right to be equal.
• ❓ The function f(x) in the given problem is continuous at x=0.
• ☺️ The cases for x less than or equal to 0 and x greater than 0 are considered separately to evaluate the continuity of f(x) at x=0.
• ☺️ The limit of sine(x) as x approaches 0 is 0, which simplifies the calculations.

Questions & Answers

Q: What does it mean for a function to be continuous at a specific point?

For a function to be continuous at a specific point, the limit as x approaches that point from the left must be equal to the limit from the right, and both limits must be equal to the value of the function at that point.

Q: Why is it important for a function to be continuous at x=0?

Continuity at x=0 ensures that there are no gaps or jumps in the graph of the function. It guarantees a smooth, connected curve without any abrupt changes in value.

Q: How is the limit of f(x) as x approaches 0 from the left calculated?

The limit as x approaches 0 from the left of f(x) is calculated by substituting the value of x=0 into the expression for f(x) when x is less than or equal to 0. In this case, it is 1 minus 2 sine of x, which simplifies to 1.

Q: What is the value of the limit of f(x) as x approaches 0 from the right?

The value of the limit as x approaches 0 from the right of f(x) is obtained by substituting the value of x=0 into the expression for f(x) when x is greater than 0. In this case, it is e to the negative 4x, which simplifies to 1.

Summary & Key Takeaways

• Problem: Determine if function f(x) is continuous at x=0.

• In order to be continuous, the limit as x approaches 0 from the left and the right should be equal to the value of the function at x=0.

• The function f(x) is continuous at x=0 because the limit from the left, the limit from the right, and the function value all equal 1.