Problem 26 - Brief Applied Calculus

Question

Find the following limits without using a graphing calculator or making tables. $$ \lim _{x \rightarrow-4} \frac{x^{2}+9 x+20}{x+4} $$

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Identify Indeterminacy

Before we proceed with finding this limit, check whether substitution of \(x = -4\) directly results in an indeterminate form. By substituting \(-4\) into the denominator and numerator, we get \((x + 4) = (-4 + 4) = 0\) and the numerator becomes \((x^2 + 9x + 20) = (-4)^2 + 9(-4) + 20 = 16 - 36 + 20 = 0\). Hence, direct substitution leads to \(\frac{0}{0}\), indicating an indeterminate form.

Step 2: Factor the Numerator

Since the direct substitution leads to an indeterminate form, we will attempt to factor the numerator \(x^2 + 9x + 20\). Look for two numbers that multiply to \(20\) and add up to \(9\). The numbers \(4\) and \(5\) work: \((x + 4)(x + 5)\). Thus, \(x^2 + 9x + 20 = (x + 4)(x + 5)\).

Step 3: Simplify the Expression

Substitute the factored form of the numerator into the original expression. We have: \[ \frac{(x+4)(x+5)}{x+4} \]. Cancel out the common factor \((x + 4)\) in the numerator and the denominator, simplifying the expression to \(x + 5\).

Step 4: Evaluate the Limit

Now that the expression is simplified, substitute \(x = -4\) back into the expression \(x + 5\). Substitute to find \(-4 + 5 = 1\). Thus, the limit is \(1\).

Key Concepts

Indeterminate Form

When evaluating limits in calculus, one can often encounter results that aren't immediately clear, known as 'indeterminate forms'. A common indeterminate form is \( \frac{0}{0} \), which doesn't provide enough information to determine the limit directly.
This occurs when both the numerator and denominator approach zero.

• Substitution into the expression gives \( 0/0 \).
• This does not tell you the value of the limit, indicating further analysis is needed.

Understanding that \( \frac{0}{0} \) is an indeterminate form signals that more algebraic manipulation, such as factoring or simplification, is required to resolve the limit. It's a pointer that the function might need to be expressed differently to remove the problematic zero division.

Factoring Polynomials

Factoring is a valuable technique for simplifying polynomial expressions, which is often necessary when dealing with indeterminate forms. In the given exercise, we are tasked with factoring the quadratic expression \( x^2 + 9x + 20 \).
Finding factors involves identifying two numbers that multiply to the constant term (in this case, 20) while adding up to the linear coefficient (9 in this example).

• Identify numbers: For 20 and 9, these numbers are 4 and 5.
• Express as factors: Rewriting gives \( (x + 4)(x + 5) \).

Factoring allows us to break the quadratic into components, often revealing common factors that can cancel with those in a denominator, simplifying the expression and helping find limits.

Simplifying Expressions

Simplifying expressions is a crucial step in resolving limits, especially when indeterminate forms like \( \frac{0}{0} \) appear. After factoring the expression, we may simplify by cancelling out common factors.
Consider the expression \( \frac{(x+4)(x+5)}{x+4} \):

• Cancel the common factor: Here, \( (x+4) \) appears in both the numerator and the denominator.
• The expression reduces to \( x+5 \).

By simplifying, the function's original form that caused \( \frac{0}{0} \) is removed. This simplification provides a direct form to substitute for \( x \) without concerns of division by zero, making the evaluation of the limit straightforward by plugging \( x = -4 \) into the simplified expression.