Problem 1 - Contemporary Precalculus

Question

Use the table feature of your calculator to find the limit. $$\lim _{x \rightarrow-1} \frac{x^{6}-1}{x^{4}-1}$$

Step-by-Step Solution

Answer

Question: Determine the limit of the function as x approaches -1: $\lim_{x \rightarrow-1} \frac{x^6 - 1}{x^4 - 1}$. Answer: The limit as x approaches -1 does not exist, or in other words, the limit is infinity: $\lim_{x \rightarrow-1} \frac{x^6 - 1}{x^4 - 1} = \infty$.

Step 1: Factor the numerator and denominator polynomials

In this step, we will factor the numerator polynomial (\(x^6 - 1\)) and the denominator polynomial (\(x^4 - 1\)) completely. $$x^6 - 1 = (x^2 + 1)(x^4 - x^2 + 1)$$ $$x^4 - 1 = (x^2 + 1)(x^2 - 1)$$ Now, our expression becomes: $$\lim _{x \rightarrow-1} \frac{(x^2 + 1)(x^4 - x^2 + 1)}{(x^2 + 1)(x^2 - 1)}$$

Step 2: Simplify the expression

In this step, we simplify the expression by canceling the common factor \(x^2 + 1\) in both the numerator and the denominator: $$\lim _{x \rightarrow-1} \frac{x^4 - x^2 + 1}{x^2 - 1}$$

Step 3: Evaluate the limit

Now we will substitute x = -1 into the simplified expression and evaluate the limit: $$\lim _{x \rightarrow-1} \frac{(-1)^4 - (-1)^2 + 1}{(-1)^2 - 1} = \frac{1 - 1 + 1}{1 - 1}$$ At this point, we have a 0 in the denominator which means the expression is undefined. In order to find the limit using the table feature of the calculator, follow these steps:

Step 4: Using the table feature of your calculator

(Note that specific calculator procedures may vary depending on the model) 1. Enter the simplified function \(\frac{x^4 - x^2 + 1}{x^2 - 1}\) in your calculator's function input. 2. Open the table settings and set the "Independent" variable to "Ask." 3. Go to the table and type in several x-values close to -1 (e.g., -1.01, -1.001, -1.0001) from both smaller and larger values. 4. Observe the output values and see if they approach a specific number or if they appear to be unbounded. As you follow these steps, you will find that the output values approach an unbounded value (infinity). This implies that the function does not have a finite limit as x approaches -1.

Step 5: Conclusion

Therefore, the limit of the given function as x approaches -1 does not exist, or in other words, the limit is infinity: $$\lim _{x \rightarrow-1} \frac{x^{6}-1}{x^{4}-1} = \infty$$

Key Concepts

Polynomial Factoring

Factoring is an essential skill in algebra and calculus that involves breaking down a complex expression into the product of simpler factors. It's like unravelling a complex puzzle.
In the exercise provided, we factor the polynomials in both the numerator and the denominator to simplify the original expression. Let's take a closer look:

• The numerator, which is a sixth-degree polynomial, is factored as:\[x^6 - 1 = (x^2 + 1)(x^4 - x^2 + 1)\]
• The denominator, a fourth-degree polynomial, is factored as:\[x^4 - 1 = (x^2 + 1)(x^2 - 1)\]

This process helps in simplifying the limit evaluation later. By identifying and removing shared factors, such as \(x^2 + 1\), we simplify the expression, making it easier to analyze.

Simplifying Expressions

Once the expression is factored, the next step involves simplifying it by cancelling out any common factors between the numerator and denominator. In our case:

• The factor \(x^2 + 1\) is present in both parts. By cancelling it, the expression becomes:\[ \frac{x^4 - x^2 + 1}{x^2 - 1} \]

Simplifying expressions is crucial because it reduces the complexity and gives a clearer picture of what happens as the variable approaches a specified value. Simplification helps alleviate potential zero-in-the-denominator problems, although in this example, further investigation showed undefined limits remain.
This prompts further analysis through other methods, such as using a calculator.

Using Calculators for Limits

While theoretical simplification often suffices for calculating limits, sometimes using a calculator's table feature can offer additional insights, especially when the usual algebraic methods hit a roadblock.
By entering the simplified function into a calculator:

• Set the calculator to show a table of values.
• Choose values for \(x\) approaching -1 from both sides, such as -1.01 and -0.99.
• Observe how the output values behave as \(x\) nears -1.

This table approach underscores how these calculations help identify trends or behaviors in the function, such as unbounded growth leading to conclusions about non-existent finite limits. This method is particularly useful when the expression results in an undefined form under simple substitution, marking a crucial step towards a deeper understanding of limits.