Problem 21 - Calculus and its applications

Question

The initial substitution of \(x=a\) yields the form \(0 / 0 .\) Look for ways to simplify the function algebraically, or use a table or graph to determine the limit. When necessary, state that the limit does not exist. $$ \lim _{x \rightarrow-2} \frac{x^{2}-2 x-8}{x^{2}-4} $$

Step-by-Step Solution

Answer

The limit is \( \frac{3}{2} \).

Step 1: Substitute to Check the Indeterminate Form

First, substitute the value of \( x = -2 \) into the function to check if it results in an indeterminate form. After substitution, we get:\[\frac{(-2)^2 - 2(-2) - 8}{(-2)^2 - 4} = \frac{4 + 4 - 8}{4 - 4} = \frac{0}{0}\]Since this results in \( \frac{0}{0} \), which is indeterminate, we'll need to simplify the expression.

Step 2: Simplify the Numerator

Factor the numerator \( x^2 - 2x - 8 \). We look for two numbers that multiply to \(-8\) and add up to \(-2\). These numbers are \(-4\) and \(+2\), so we can write:\[x^2 - 2x - 8 = (x - 4)(x + 2)\]This gives us a factored form of the numerator.

Step 3: Simplify the Denominator

Factor the denominator \( x^2 - 4 \) using the difference of squares method. The expression \( x^2 - 4 \) can be written as:\[x^2 - 4 = (x - 2)(x + 2)\]Now both the numerator and the denominator are in their factored forms.

Step 4: Cancel the Common Factor

Notice that \( x + 2 \) is a common factor in both the numerator and the denominator:\[\frac{(x - 4)(x + 2)}{(x - 2)(x + 2)}\]Cancel out the \( x + 2 \) terms:\[\frac{x - 4}{x - 2}\]This simplification removes the indeterminate form, allowing us to evaluate the limit.

Step 5: Evaluate the Simplified Limit

With the simplified function \( \frac{x - 4}{x - 2} \), substitute \( x = -2 \):\[\frac{-2 - 4}{-2 - 2} = \frac{-6}{-4} = \frac{3}{2}\]Thus, the limit of the original function as \( x \) approaches \(-2\) is \( \frac{3}{2} \).

Key Concepts

Indeterminate Forms

When evaluating limits, an indeterminate form often signals a need for further analysis. The most common indeterminate form is \( \frac{0}{0} \), which arises when substitution into a limit results in both the numerator and the denominator being zero.
This signals that the function’s behavior is not immediately clear near the point of evaluation. It does not allow for direct assertion of the limit's value.To deal with indeterminate forms:

• First, substitute the value into the function. If the result is \( \frac{0}{0} \), further action is needed.
• Look for algebraic methods to simplify. This often involves factoring or using algebraic manipulations.
• Sometimes, graphical or numerical approaches are useful to predict if the limit exists.

These strategies are crucial in determining the limit or, in some cases, confirming the nonexistence of the limit.

Factoring Polynomials

Factoring is a key technique used to simplify expressions involving polynomials. It involves expressing a polynomial as a product of its factors, which are simpler polynomials. This process simplifies complex expressions and makes calculations, such as finding limits, manageable.
To factor a quadratic polynomial, like \( x^2 - 2x - 8 \):

• Identify two numbers that multiply to the constant term (-8) and add to the linear coefficient (-2).
• For \( x^2 - 2x - 8 \), the numbers -4 and 2 satisfy both conditions.
• Rewrite the quadratic as a product: \((x - 4)(x + 2)\).

Factoring polynomials is crucial in revealing common factors between the numerator and denominator, enabling the simplification of rational expressions.

Simplifying Rational Expressions

Simplifying rational expressions involves reducing them to their simplest form by eliminating common factors. This process often follows once terms in the expressions are factored.
Here's how to simplify a rational expression:

• After factoring both the numerator and the denominator, identify common factors.
• Cancel the common factors, making sure no division by zero occurs.
• For example, with \( \frac{(x - 4)(x + 2)}{(x - 2)(x + 2)} \), the factor \( x + 2 \) appears in both the numerator and the denominator.
• Cancelling \( x + 2 \) yields \( \frac{x - 4}{x - 2} \).

This simplification solves the indeterminate form and allows for the evaluation of the limit. Simplifying rational expressions is essential for understanding the underlying behavior of functions near specific points.