Skills Review for Comparison Tests

Learning Outcomes

  • Simplifying expressions using the Quotient Property of Exponents
  • Calculate the limit of a function as 𝑥 increases or decreases without bound
  • Recognize when to apply L’Hôpital’s rule

In the Comparison Tests section, we will explore some more methods that can be used to determine whether an infinite series diverges or converges. Here we will review the quotient rule for exponents, how to take limits at infinity, and L’Hopital’s Rule.

The Quotient Rule for Exponents

A General Note: The Quotient Rule of Exponents

For any real number [latex]a[/latex] and natural numbers [latex]m[/latex] and [latex]n[/latex], such that [latex]m>n[/latex], the quotient rule of exponents states that

[latex]\dfrac{{a}^{m}}{{a}^{n}}={a}^{m-n}[/latex]

Example: Using the Quotient Rule

Write each of the following products with a single base. Do not simplify further.

  1. [latex]\dfrac{{\left(-2\right)}^{14}}{{\left(-2\right)}^{9}}[/latex]
  2. [latex]\dfrac{{t}^{23}}{{t}^{15}}[/latex]
  3. [latex]\dfrac{{\left(z\sqrt{2}\right)}^{5}}{z\sqrt{2}}[/latex]

Try It

Take Limits at Infinity

(see Module 5, Skills Review for Sequences.)

Infinite Limits at Infinity

(see Module 5, Skills Review for Sequences.)

Apply L’Hôpital’s Rule

(see Module 5, Skills Review for Sequences.)