Problem 85 - Precalculus with Limits

Question

TRUE OR FALSE? In Exercises 85 and 86, determine whether the statement is true or false. Justify your answer. When your attempt to find the limit of a rational function yields the indeterminate form \(\frac{0}{0}\) the rational function's numerator and denominator have a common factor.

Step-by-Step Solution

Answer

The statement is true. Whenever the limit of a rational function yields an indeterminate form of \(\frac{0}{0}\), it means both the numerator and denominator are zero at the limit point and hence, a common factor exists. You can confirm this by cancelling out the common factor, simplifying the function, and reapplying the limit.

Step 1: Understanding Rational Functions and Limits

A rational function is a function of the form \(\frac{f(x)}{g(x)}\), where f(x) and g(x) are polynomial functions. The limit of a function at a point is the value that the function approaches as it gets infinitesimally close to that point. In the context of rational functions, it is imperative to understand that the zero in the denominator often points to spinning out of control or indeterminate situation.

Step 2: Introduction to Indeterminate Form: \(\frac{0}{0}\)

The term \(\frac{0}{0}\) is referred to as indeterminate form because it's not possible to determine its exact value given that division by zero is undefined in mathematics. However, such form often tells you there's more to the story and further analysis might be needed, often employing L'Hopital's Rule.

Step 3: Statement Verification

If an attempt to find the limit of a function yields the indeterminate form \(\frac{0}{0}\), this implies that at the limit point, both the numerator and the denominator are zero. Given the polynomial characteristics of the numerator and denominator in a rational function, if both become zero at the same limit point, they must have a common factor. When we factorize the common factor, we could apply the limit and this process is known as the cancelation of common factor.

Key Concepts

Indeterminate Form

When evaluating limits of rational functions, you might encounter the expression \(\frac{0}{0}\). This is known as an indeterminate form. The term "indeterminate" indicates that it does not have a specific value or conclusion directly from its appearance.

• Indeterminate forms are significant in calculus because they signal that simple substitution into the function does not work and more analytical methods are required.
• It's important to remember that \(\frac{0}{0}\) differs from a simple undefined quantity. It implies the need for further examination, potentially revealing more about the behavior of the function around the point of interest.

Understanding the nature of indeterminate forms is crucial, as they often guide us to apply techniques like L'Hopital's Rule or algebraic manipulation to resolve the limit.

L'Hopital's Rule

Once you've identified that you have an indeterminate form such as \(\frac{0}{0}\), L'Hopital's Rule can be a useful method to evaluate the limit.

• L'Hopital's Rule states that if you have a limit that results in \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), you can differentiate the numerator and the denominator separately and then take the limit of their derivatives.
• This approach often simplifies the problem, making it possible to resolve limits that initially appear undefined or indeterminate.
• Remember, L'Hopital's Rule applies iteratively. If differentiating once still results in an indeterminate form, you might need to differentiate again until a determinate form is achieved.

Employing L'Hopital's Rule effectively transforms complex limit problems into simpler ones that are easier to solve.

Polynomial Functions

Rational functions are built from polynomial functions in both the numerator and the denominator. Understanding polynomial functions is key to handling rational function limits, especially when encountering indeterminate forms.

• Polynomial functions are expressions composed of variables and coefficients, involving operations of addition, subtraction, and non-negative integer exponents on the variables.
• These functions are the building blocks for rational functions, often setting the stage for indeterminate forms like \(\frac{0}{0}\), depending on their roots and behavior near specific points.
• Common factors between the numerator and the denominator can lead to indeterminate forms. Factoring allows for the simplification of these rational functions, sometimes cancelling terms that contribute to the indeterminate behavior.

Understanding how to manipulate and simplify polynomial expressions is a powerful tool when seeking to find the accurate limit of a rational function.