Problem 17 - Calculus and its applications

Question

Use the Theorem on Limits of Rational Functions to find each limit. When necessary, state that the limit does not exist. $$ \lim _{x \rightarrow 3} \frac{x^{2}-25}{x^{2}-5} $$

Step-by-Step Solution

Answer

The limit is \(-4\).

Step 1: Identify the Function Form

The given function is a rational function, \(f(x) = \frac{x^{2}-25}{x^{2}-5}\). A rational function is a fraction of two polynomials.

Step 2: Identify the Limit Point

We need to find the limit as \(x\) approaches 3. Hence, our task is to evaluate \(\lim_{x \to 3} \frac{x^{2}-25}{x^{2}-5}\).

Step 3: Substitute the Limit Point

Substitute \(x = 3\) directly into the function: \(\frac{3^{2} - 25}{3^{2} - 5}\). This gives us \(\frac{9 - 25}{9 - 5} = \frac{-16}{4} = -4\).

Step 4: Verify Conditions for Direct Substitution

Since both the numerator and denominator evaluated at \(x = 3\) do not yield 0, direct substitution is valid and yields the limit.

Key Concepts

Rational Functions

Understanding rational functions is crucial when dealing with limits, especially as they often appear in calculus problems. A rational function is a type of mathematical expression representing the ratio of two polynomials. It is written in the form \( f(x) = \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomial functions.

Some important characteristics of rational functions include:

• They may have holes or vertical asymptotes where the denominator equals zero.
• Rational functions are not defined for values of \( x \) that make \( Q(x) = 0 \).

In cases where we're finding limits, determining if a rational function is defined at a point can reveal if direct substitution is a suitable method.

In our specific example, the function is \( f(x) = \frac{x^{2}-25}{x^{2}-5} \). Here, both the numerator, \( x^2 - 25 \), and the denominator, \( x^2 - 5 \), are polynomials, confirming that this is indeed a rational function.

Direct Substitution

Direct substitution is a straightforward method used to find limits, especially effective with rational functions that are defined at the limit point. This method involves plugging the value you're approaching directly into the function. When both the numerator and denominator are non-zero after substitution, it's typically an indication that unsubstituted evaluation will yield the limit without issue.

In solving the given exercise, we evaluated the function \( \frac{x^{2}-25}{x^{2}-5} \) at \( x = 3 \). When substituting \( x = 3 \), it resulted in:

• Numerator: \( 3^2 - 25 = 9 - 25 = -16 \)
• Denominator: \( 3^2 - 5 = 9 - 5 = 4 \)

Both results are non-zero, which confirms that the limit exists and can be directly calculated. The resulting calculation gave us \( \frac{-16}{4} \) which simplifies to \(-4\).

This shows how powerful direct substitution can be in simplifying the process of finding limits.

Polynomial Functions

Polynomial functions are foundational in determining the behavior of rational functions. Understanding them is critical when analyzing the components of rational functions like those involved in our limit problem.

A polynomial function is an expression of the form \( P(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \). Key features include:

• They are continuous and smooth, making them easier to work with in calculus.
• The degree of a polynomial is the highest power of \( x \) present, influencing the graph's end behavior.

In the example \( f(x) = \frac{x^{2}-25}{x^{2}-5} \), both the numerator \( x^2 - 25 \) and the denominator \( x^2 - 5 \) are polynomials of degree 2:

• The numerator can be rewritten as \( (x-5)(x+5) \) using the difference of squares.
• The denominator \( x^2 - 5 \) is already in a simple polynomial form.

Recognizing these forms enables a deeper understanding of how these functions behave as \( x \) approaches a value like 3, aiding in finding the limits straightforwardly.