Problem 33 - Calculus: Early Transcendentals

Question

Evaluate limit and justify your answer. $$\lim _{x \rightarrow 4} \sqrt{\frac{x^{3}-2 x^{2}-8 x}{x-4}}$$

Step-by-Step Solution

Answer

Answer: The limit as x approaches 4 of the given expression is $\sqrt{24}$.

Step 1: Factor the numerator

First, let's factor the numerator to simplify the fraction inside the square root: $$x^{3}-2x^{2}-8x = x(x^{2}-2x-8) = x(x-4)(x+2)$$

Step 2: Simplify the expression inside the square root

Now we can simplify the expression inside the square root by canceling out the common factor in the numerator and the denominator: $$\frac{x(x-4)(x+2)}{x-4} = x(x+2)$$

Step 3: Check the function for non-negativity

Before we proceed, we need to make sure that the function inside the square root is non-negative for all x values around 4, otherwise the limit will not exist. Since x(x+2) is non-negative for all x values greater than or equal to -2, and 4 is greater than -2, we can find the limit of the simplified expression.

Step 4: Evaluate the limit of the simplified expression

Now, let's find the limit of the square root of the simplified expression: $$\lim _{x \rightarrow 4} \sqrt{x(x+2)}$$ We can use the limit properties and substitution method to find the limit: $$\lim _{x \rightarrow 4} \sqrt{x(x+2)} =\sqrt{4(4+2)}=\sqrt{4(6)}=\sqrt{24}$$ So, the limit as x approaches 4 of the given expression is: $$\lim _{x \rightarrow 4} \sqrt{\frac{x^{3}-2 x^{2}-8 x}{x-4}} = \sqrt{24}$$

Key Concepts

Factoring Polynomial Expressions

Factoring polynomial expressions is a crucial step in solving calculus limits problems, particularly when working with expressions that appear complex. The goal of factoring is to break down a polynomial expression into simpler components, known as factors, which are easier to work with. In our exercise, the expression we worked with was \(x^{3} - 2x^{2} - 8x\).

To factor this expression, we look for common factors and apply basic algebraic identities. First, we can see that each term in the polynomial shares a common factor of \(x\). By factoring \(x\) out of the entire expression, we transform it into \(x(x^{2} - 2x - 8)\).

Next, we focus on the quadratic polynomial \(x^{2} - 2x - 8\). We need to find two numbers that multiply to -8 (the constant term) and add to -2 (the coefficient of \(x\)). These numbers are -4 and 2. Therefore, the quadratic can be factored as \((x-4)(x+2)\).

• Recall: Always check for common factors first.
• Look for patterns or identities like difference of squares or perfect squares.
• Verify your factors by multiplying them back together.

After factoring, the original expression simplifies to \(x(x-4)(x+2)\), making it easier to handle in subsequent steps.

Simplifying Rational Expressions

Simplifying rational expressions means reducing them to their simplest form. This involves canceling out common factors in the numerator and the denominator. Simplification is important because it can remove discontinuities, making it easier to evaluate limits.

In our exercise, once the polynomial was factored, we obtained the expression \( \frac{x(x-4)(x+2)}{x-4} \). We noticed a common factor of \(x-4\) in both the numerator and the denominator. By canceling out this common factor, we simplify the expression to \(x(x+2)\).

• Always factor before simplifying.
• Check for common factors in both the numerator and denominator.
• Be cautious about domain restrictions; cancelled factors affect the domain.

By cancelling \(x-4\), the expression remains valid because we are specifically looking at a limit, which allows us to disregard this factor provided it doesn't cause additional discontinuities in the neighborhood of the limit point.

Substitution in Limits

Substitution is a technique used to evaluate limits by plugging in the value that \(x\) approaches into the simplified expression. Once the rational expression is simplified, substitution can help check if the expression results in an indeterminate form or a specific value.

In solving our exercise, after simplifying the expression to \(x(x+2)\), we focus on evaluating the limit \(\lim_{x \to 4} \sqrt{x(x+2)}\). Using substitution, we replace \(x\) with 4, yielding \(\sqrt{4(4+2)} = \sqrt{24}\). By directly substituting \(x = 4\), we verify that the function value does not create an indeterminate form or undefined behavior around this point.

• Substitute after simplifying to avoid indeterminate forms like \( \frac{0}{0} \).
• Always ensure the function is defined in the neighborhood of the point of limit.
• Verify that the entire substituted value is valid before concluding the evaluation.

This method pinpoints the behavior of the function as \(x\) approaches a specific value, confirming the limit of the original complex expression as \(\sqrt{24}\).