Problem 14 - Thomas Calculus

Question

Find the limits in Exercises \(11-22\) $$\lim _{x \rightarrow-2}\left(x^{3}-2 x^{2}+4 x+8\right)$$

Step-by-Step Solution

Answer

The limit is -16.

Step 1: Substitution Check

First, we attempt to find the limit by direct substitution of \( x = -2 \) into the polynomial \( x^3 - 2x^2 + 4x + 8 \).

Step 2: Evaluating the Expression

Substitute \( x = -2 \) into the expression: \[ (-2)^3 - 2(-2)^2 + 4(-2) + 8. \]

Step 3: Calculate Each Term

Calculate each term separately:- \( (-2)^3 = -8 \).- \( 2(-2)^2 = 2(4) = 8 \).- \( 4(-2) = -8 \).- \( 8 = 8 \).

Step 4: Combine the Results

Add the calculated terms together:\[ -8 - 8 + (-8) + 8 = -16. \]

Step 5: Conclusion: Determine the Limit

The limit of the polynomial function as \( x \) approaches \( -2 \) is \( -16 \).

Key Concepts

Direct Substitution

Direct substitution is a straightforward method used to evaluate limits, particularly when functions are continuous at the point of interest. It's a simple process where you replace the variable with the number it approaches. In problems involving limits, this technique can be incredibly helpful.

For example, when we are asked to find the limit of a function as \( x \) approaches a certain value, our first instinct might be to substitute the value directly into the function. If the function remains well-defined (meaning there are no indeterminate forms like \( \frac{0}{0} \)), the result is simply the limit.

• This method works smoothly for polynomial functions because they are continuous everywhere on their domain.
• Direct substitution is usually the first step in limit evaluation, providing quick results for many functions.

However, if directly substituting the value leads to an undefined expression, alternative methods, such as factoring or using L'Hopital's Rule, might be necessary.

Polynomial Function

Polynomial functions are expressions that involve variables raised to whole number exponents with coefficients. These are among the simplest and most common functions in mathematics.

The general form of a polynomial function is \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 \), where \( a_n, a_{n-1}, \ldots, a_0 \) are coefficients and \( n \) is a non-negative integer representing the degree of the polynomial.

• Polynomial functions are defined and continuous everywhere, meaning you can plug any value of \( x \) into a polynomial and get a result.
• This property makes them perfect candidates for direct substitution when evaluating limits.

In the exercise provided, the polynomial \( x^3 - 2x^2 + 4x + 8 \) is evaluated by substituting a value directly to find the limit. Because polynomials behave so predictably, they are excellent for practicing limit problems.

Limit Evaluation

Evaluating limits is a fundamental technique in calculus for understanding how a function behaves as its input approaches a particular value. Limits help us find behavior trends near critical points, rather than actual outputs.

Here's how the limit of a function is generally evaluated:

• First, attempt to substitute the approaching value directly into the function. If it results in a defined output, then that is your limit.
• If direct substitution results in an undefined form, further methods are applied.

In the case of the given problem, we were tasked with finding the limit as \( x \) approaches \(-2\) for a polynomial function. By using direct substitution, we placed \( x = -2 \) into \( x^3 - 2x^2 + 4x + 8 \) and calculated term by term, leading to the limit being \(-16\).

This exercise highlights how polynomial functions often provide straightforward scenarios for demonstrating basic limit concepts and the power of substitution as a valid computational tool.