Limit at a point of discontinuity

TL;DR

The limit of the function f(x) as x approaches three does not exist, as it approaches different values from the left and right sides.

Transcript

Let's say that f(x) is equal to the absolute value of x minus three over x minus three and what I'm curious about is the limit of f(x) as x approaches three and just from an inspection you can see that the function is not defined when x is equal to three - you get zero over zero: it's not defined So to answer this question let's try to re-write the... Read More

Key Insights

• ☺️ The function f(x) is undefined at x = 3 due to division by zero.
• ☺️ When x is greater than three, the function simplifies to f(x) = 1.
• ☺️ When x is less than three, the function simplifies to f(x) = -1.
• 👈 The limit of f(x) as x approaches three from the left side is -1.
• 👉 The limit of f(x) as x approaches three from the right side is 1.
• ☺️ The limit of the function as x approaches three does not exist.
• 😥 The concept of limits helps us analyze the behavior of functions near specific points.

Questions & Answers

Q: Why is the function not defined at x = 3?

The function f(x) is not defined at x = 3 because it results in a division by zero, which is undefined.

Q: What is the limit of f(x) as x approaches three from the left side?

The limit of f(x) as x approaches three from the left side is -1, as the function approaches -1 as x gets closer to 3 from values less than 3.

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

The limit of f(x) as x approaches three from the right side is 1, as the function approaches 1 as x gets closer to 3 from values greater than 3.

Q: What does it mean when the limit of a function does not exist?

When the limit of a function does not exist, it means that the function approaches different values from the left and right sides, indicating that there is no single value that the function is converging to.

Summary & Key Takeaways

• The function f(x) is not defined at x = 3, as it results in a division by zero.

• When x is greater than three, the function simplifies to f(x) = 1.

• When x is less than three, the function simplifies to f(x) = -1.