Problem 23 - 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. $$ \lim _{x \rightarrow 2} \frac{3 x^{2}+x-14}{x^{2}-4} $$

Step-by-Step Solution

Answer

The limit does not exist, as the expression simplifies to division by zero.

Step 1: Substitution Check

First, substitute \(x = 2\) into the function to check if it yields an indeterminate form. Substituting into the numerator: \(3(2)^2 + 2 - 14 = 12 + 2 - 14 = 0\). Substituting into the denominator: \((2)^2 - 4 = 4 - 4 = 0\). Thus, the limit initially takes the form \( \frac{0}{0} \), indicating we must simplify further.

Step 2: Factor the Denominator

Recognize that the denominator \(x^2 - 4\) is a difference of squares and can be factored as: \(x^2 - 4 = (x-2)(x+2)\).

Step 3: Factor the Numerator

Attempt to factor the numerator \(3x^2 + x - 14\). After testing for roots and using methods such as trial and error or the quadratic formula, we find the factors: \(3x^2 + x - 14 = (3x - 7)(x + 2)\).

Step 4: Simplify the Expression

After factoring, the expression becomes: \[\frac{(3x - 7)(x + 2)}{(x-2)(x+2)}\]. Observe that \((x+2)\) is present in both the numerator and the denominator, allowing us to cancel it out. This simplifies the expression to: \[\frac{3x - 7}{x-2}\].

Step 5: Re-evaluate the Limit

Substitute \(x = 2\) again into the simplified expression: \(\frac{3(2) - 7}{2 - 2} = \frac{6 - 7}{0} = \frac{-1}{0}\). Since this results in division by zero, the limit is undefined.

Key Concepts

Indeterminate Forms

When evaluating limits in calculus, you might often encounter an expression that results in the form \( \frac{0}{0} \). This is known as an indeterminate form. This form tells you that just substituting the value directly into the function won’t work to find the limit. But don't worry! Understanding indeterminate forms helps us to recognize when we need to take extra steps to simplify or analyze the function further. Key methods include algebraic simplification, using graphs, or analyzing behavior close to the point of interest. The goal: find the value the function approaches as it gets close to a specific point. Indeterminate forms are like road signs saying "Keep Going! Find Another Way!" to determine the limit.

Polynomial Factorization

Polynomial factorization is a technique used to rewrite a polynomial as a product of its factors. These factors are usually simpler polynomials. This process is crucial in simplifying expressions that have polynomials in both the numerator and denominator. By factoring, you can often cancel out common factors which allows you to simplify the expression significantly. For instance, if you have a polynomial like \( x^2 - 4 \), recognize it as a difference of squares, which can be broken down into \((x-2)(x+2)\). In our example problem, this was key for simplifying the limit expression. By uncovering hidden factors, you can solve problems that initially appears complicated.

Limit Simplification

Limit simplification is about turning complicated algebraic expressions into simpler ones to make studying behavior around a limit clearer. Often the original function directly substituted produces an indeterminate form like \( \frac{0}{0} \). The act of simplifying involves:

• Factoring polynomials
• Cancelling out common terms
• Reassessing the limit with substituted simpler terms.

In this problem, factorization was used to remove common terms and facilitate the limit process. Once you've reduced the expression, retry substitution to find the limit clearly, or to determine if it does not exist.

Difference of Squares

Sometimes, simplifying polynomial expressions involves recognizing certain patterns, such as the difference of squares. The difference of squares is a specific factorization pattern for two terms that are squares separated by subtraction. It appears as \( a^2 - b^2 \) and can be rewritten as \((a-b)(a+b)\). In our task, \( x^2 - 4 \) fits this pattern perfectly, as \( x^2 - 4 \) is equivalent to \((x-2)(x+2)\). Use this method to break down complex rational expressions in limits, which may help simplify and resolve indeterminate forms. By spotting this pattern, you can simplify both the setup and solution of many limits problems.