What Are Removable and Nonremovable Discontinuities?

TL;DR

Removable discontinuities in rational functions are represented by holes, which occur when factors in the numerator and denominator cancel out. Nonremovable discontinuities correspond to vertical asymptotes, occurring when the denominator equals zero and cannot be canceled. In essence, holes are removable, while vertical asymptotes are not.

Transcript

the question wants to know you know what are the removable and non removable Descanso fill-in-the-blank removable and non removable discontinuities what are they right they're removable and not removable so solution so something happens to the bottom of this fraction what can you do on the bottom yeah what's that called factors I hope if I did it r... Read More

Key Insights

• 🕳️ Removable discontinuities occur when there are cancellations in a rational function, resulting in holes in the graph.
• 🕳️ Holes in rational functions are always removable.
• 🚦 Non-removable discontinuities in rational functions are vertical asymptotes, occurring when the denominator becomes zero.
• 🚦 Vertical asymptotes are always non-removable.
• #️⃣ Rational functions are polynomials over polynomials, with whole number coefficients.
• 🍉 Removable discontinuities can be identified by cancelled terms in the function.
• 😫 Vertical asymptotes can be found by setting the denominator of the rational function equal to zero.

Questions & Answers

Q: What are removable and non-removable discontinuities?

Removable discontinuities are holes in a rational function, while non-removable discontinuities are vertical asymptotes.

Q: How can one identify a removable discontinuity in a rational function?

If there is a cancellation that results in a hole, it is a removable discontinuity. It is represented as a fraction where the numerator and denominator have a common factor.

Q: Are all holes in rational functions removable?

Yes, all holes in rational functions are removable. This means that they can be filled by simplifying the function.

Q: What determines a non-removable discontinuity in a rational function?

Non-removable discontinuities in rational functions are vertical asymptotes. They occur where the denominator of the function becomes zero.

Summary & Key Takeaways

• Removable discontinuities refer to holes in a rational function, while non-removable discontinuities refer to vertical asymptotes.

• Holes in rational functions are always removable, while vertical asymptotes are always non-removable.

• Rational functions are polynomials over polynomials, with powers of x and whole number coefficients.