Problem 15 - Precalculus: Mathematics for Calculus

Question

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). $$\lim _{x \rightarrow 1}\left(\frac{x^{4}+x^{2}-6}{x^{4}+2 x+3}\right)^{2}$$

Step-by-Step Solution

Answer

The limit is \( \frac{4}{9} \).

Step 1: Identify the Expression to Evaluate

We want to evaluate the limit of the following expression as \( x \) approaches 1: \( \lim _{x \rightarrow 1}\left(\frac{x^{4}+x^{2}-6}{x^{4}+2 x+3}\right)^{2} \).

Step 2: Use the Quotient Rule

First, find the limit of the quotient inside the limit expression: \( \lim _{x \rightarrow 1} \frac{x^{4} + x^{2} - 6}{x^{4} + 2x + 3} \). Applying Limit Laws, and if the limit exists for both the numerator and the denominator, then: \( \lim_{x \to a} \left( \frac{f(x)}{g(x)} \right) = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)} \) assuming \( \lim_{x \to a} g(x) eq 0 \).

Step 3: Evaluate Limits of Numerator and Denominator

For the numerator \( x^4 + x^2 - 6 \), evaluate the limit as \( x \to 1 \): \( 1^4 + 1^2 - 6 = 1 + 1 - 6 = -4 \). For the denominator \( x^4 + 2x + 3 \), evaluate the limit as \( x \to 1 \): \( 1^4 + 2(1) + 3 = 1 + 2 + 3 = 6 \).

Step 4: Substitute Limits into the Quotient

Substitute the evaluated limits into the quotient: \( \frac{-4}{6} = -\frac{2}{3} \). Thus, \( \lim_{x \to 1} \frac{x^4 + x^2 - 6}{x^4 + 2x + 3} = -\frac{2}{3} \).

Step 5: Apply the Power Rule

The original limit expression is the square of the quotient: \( \lim _{x \rightarrow 1}\left(\frac{x^{4}+x^{2}-6}{x^{4}+2 x+3}\right)^{2} \). Using the power rule of limits, \( \lim_{x \to a} f(x)^n = (\lim_{x \to a} f(x))^n \), the limit of the squared quotient becomes: \((-\frac{2}{3})^2 = \frac{4}{9} \).

Key Concepts

Quotient Rule

The Quotient Rule is a fundamental part of evaluating limits involving expressions that are fractions. When we see a limit expression that is a fraction, like \( \lim _{x \rightarrow a} \frac{f(x)}{g(x)} \), we use the Quotient Rule to break it down. The rule states that if the limits of both the numerator and the denominator exist individually, and the limit of the denominator is not zero, we can calculate the limit of the entire fraction by:

• Finding the limit of the numerator \( f(x) \) as \( x \to a \).
• Finding the limit of the denominator \( g(x) \) as \( x \to a \).
• Calculating the quotient of these limits: \( \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)} \).

It's very handy for simplifying the limit of a ratio of two functions. Always remember, this rule requires the denominator's limit to not be zero since dividing by zero is undefined.

Power Rule of Limits

The Power Rule of Limits reveals how to handle limits when dealing with expressions raised to a power. If we have an expression like \( f(x)^n \) and want to find its limit, the rule is:

• First find the limit of the base function \( f(x) \) as \( x \to a \).
• Then raise this limit to the power \( n \): \( (\lim_{x \to a} f(x))^n \).

It's quite straightforward. This rule allows us to evaluate limits more easily when the expression is raised to an exponent. Just ensure you've precisely determined the limit of the base before applying the power. It simplifies expressions, especially when you encounter them squared, cubed, or any higher powers. Always take care with negative numbers and fractional powers.

Evaluating Limits

Evaluating limits involves understanding the behavior of functions as the input approaches a specific value. Identifying the expression and understanding which Limit Law to apply is crucial. Here's a general strategy:

• Step 1: Identify what you need to evaluate and write down the limit expression.
• Step 2: Use appropriate Limit Laws like the Quotient Rule or Power Rule as the problem requires.
• Step 3: Analyze the behavior of each part of the expression — both numerator and denominator for quotients, or base for powers.
• Step 4: Substitute values and simplify to find the exact limit.

Practicing these steps with various functions helps to master the art of limit evaluation. It's not just about applying rules but understanding why they work, making challenging problems much more manageable.