Problem 14 - Calculus: Early Transcendentals

Question

Evaluate the limit, if it exists. \( \displaystyle \lim_{x \to 4}\frac{x^2 + 3x}{x^2 - x - 12} \)

Step-by-Step Solution

Answer

The limit does not exist as substitution results in division by zero.

Step 1: Determine Indeterminate Form

First, substitute the value of the limit into the function to see if it results in an indeterminate form. Substitute \(x = 4\) into the expression: \[ \frac{4^2 + 3 \times 4}{4^2 - 4 - 12} = \frac{16 + 12}{16 - 4 - 12} = \frac{28}{0} \]Since the denominator is 0 and the numerator is not 0, it is not an indeterminate form and the limit might not exist as is.

Step 2: Factor the Denominator

To see if there are common factors that can be canceled, factor the denominator. The denominator is \(x^2 - x - 12\). To factor it, look for two numbers whose product is \(-12\) and whose sum is \(-1\). These numbers are 3 and -4, so:\[ x^2 - x - 12 = (x - 4)(x + 3) \]

Step 3: Analyze the Function Behavior

Analyze the behavior of the function around \(x = 4\). Since the expression gives an undefined point at \(x = 4\) (division by zero), check the behavior approaching from the left and right of 4. Observe that as \(x\) approaches 4, the denominator \((x-4)(x+3)\) approaches zero while the behavior of the numerator remains finite.

Key Concepts

Indeterminate Form

When evaluating the limit of a function, it's important to first check whether the expression is a so-called "indeterminate form". Indeterminate forms are expressions where the limit can't be directly determined and often involve forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \).

For the problem at hand, substituting \( x = 4 \) results in

• Numerator: \( 4^2 + 3 \times 4 = 28 \)
• Denominator: \( 4^2 - 4 - 12 = 0 \)

In this case, the numerator is 28, and the denominator turns into 0. This is not one of the indeterminate forms, such as \( \frac{0}{0} \), but rather \( \frac{28}{0} \), which implies an undefined limit where the tendency of values approaching might hint at infinity or does not exist.

Understanding indeterminate forms can guide you to re-evaluate or manipulate the expression further to make sense of the limit.

Factoring

Factoring is a common technique to simplify expressions in calculus, especially when dealing with limits.

In some cases, you might encounter a zero in the denominator, which makes the expression undefined. By factoring, you can sometimes cancel out these troubling terms.

For this problem, the denominator \( x^2 - x - 12 \) can be factored as follows:

• Find two numbers that multiply to \(-12\) and add up to \(-1\).
• These numbers are 3 and -4, so the factorization becomes \( (x - 4)(x + 3) \).

Factoring helps to see if there are common components in both the numerator and denominator that allow further simplification. Although in this example, no straightforward cancellations are available that resolve the undefined point at \( x = 4 \).

Factoring, however, provides a clearer view of the expression and helps you analyze its behavior near challenging points.

Limit Behavior

Analyzing the limit behavior involves observing what happens to a function as the input value approaches a certain point.

For this problem, the limit of the expression\[\lim_{x \to 4}\frac{x^2 + 3x}{x^2 - x - 12}\]must be evaluated by seeing how it behaves as \(x\) gets close to 4.

Even though substituting the exact value \(x = 4\) results in division by zero, it's valuable to study the behavior of the numerator and denominator as \(x\) approaches 4 from both sides:

• As \( x \) approaches 4, the denominator \((x-4)(x+3)\) approaches 0.
• The numerator, however, remains finite, resulting in an expression heading towards \( \frac{finite}{0} \).

In this situation, the limit suggests that the function could head towards \( \pm \infty \) or that it does not exist. Such a behavior often requires more context or a specific side limit evaluation for a precise conclusion.

Understanding limit behavior is crucial because it reveals continuous trends and guides problem-solving for similar calculus challenges.