Removable or Nonremovable Discontinuity Example with Absolute Value

TL;DR

Identifying and determining if a discontinuity is removable or non-removable using limits.

Transcript

this problem were given a function were asked to find the discontinuity and identify if it's removable or non removable so first of all it's pretty clear where the function is not continuous it's not continuous at x equals five and the reason is it's not defined there right if you plug in five you get zero over zero which makes no sense so it's dis... Read More

Key Insights

• 🚱 Discontinuities in functions may be removable or non-removable, depending on if they can be removed to make the function continuous.
• 👈 Calculating limits as X approaches the discontinuity point helps determine if the discontinuity is removable or non-removable.
• 🤑 Removable discontinuities are often seen in rational functions, while non-removable ones are associated with vertical asymptotes.

Questions & Answers

Q: How do you identify a discontinuity in a function?

A discontinuity occurs when a function is not continuous at a specific point, often indicated when the function is not defined at that point, like x=5 in this case.

Q: What distinguishes a removable discontinuity from a non-removable one?

A removable discontinuity can be removed to make the function continuous, while a non-removable discontinuity persists, usually seen in cases of vertical asymptotes.

Q: How can one determine if a discontinuity is removable or non-removable using limits?

By calculating the limit as X approaches the point of discontinuity (C), if the limit exists, the discontinuity is removable; if it does not exist, the discontinuity is non-removable.

Q: Why is taking one-sided limits important in analyzing discontinuities?

One-sided limits, like approaching a point from the left or right, help simplify absolute value functions and determine if the function is continuous at that specific point.

Summary & Key Takeaways

• The content explains how to identify a discontinuity at x=5 in a function and determine if it's removable or non-removable.

• Removable discontinuities can be removed to make a function continuous, while non-removable discontinuities persist.

• Calculating the limit as X approaches 5 from both sides helps determine if the discontinuity is removable or non-removable.