Removable and Nonremovable Discontinuities in a Rational Function Calculus Example | Summary and Q&A

TL;DR

Learn about the difference between removable and non-removable discontinuities in rational functions.

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.

Transcript

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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.