Problem 40 - Precalculus

Question

See if a table of values suggests a limit exists for the functions and approaches indicated. \(g(x)=\frac{x^{4}-1}{x+1}\) as \(x \rightarrow-1\) from the left.

Step-by-Step Solution

Answer

The limit is 4 as \( x \to -1^- \).

Step 1: Review the Function and Approach

The given function is \( g(x) = \frac{x^4 - 1}{x + 1} \) and we need to find the limit as \( x \to -1 \) from the left. Identify that \( g(x) \) might be undefined at \( x = -1 \) because it would make denominator zero. So, we'll need to carefully analyze the function's behavior as \( x \) approaches \(-1\) from the left side.

Step 2: Simplify the Function

The expression \( x^4 - 1 \) can be factored as \((x^2 - 1)(x^2 + 1)\). Further, \( x^2 - 1 \) can be factored into \((x - 1)(x + 1)\). Thus, \( x^4 - 1 = (x - 1)(x + 1)(x^2 + 1)\). Therefore, the function simplifies to \( g(x) = \frac{(x - 1)(x + 1)(x^2 + 1)}{x + 1} \). Cancel out \( x + 1 \) in the numerator and the denominator, yielding \( g(x) = (x - 1)(x^2 + 1) \) for \( x eq -1 \).

Step 3: Construct a Table of Values

Select values of \( x \) approaching \(-1\) from the left, such as \(-1.1, -1.01, -1.001\). Compute \( g(x) = (x - 1)(x^2 + 1) \) for each of these values to see the trend. For instance, calculate \( g(-1.1) \), \( g(-1.01) \), and \( g(-1.001) \).

Step 4: Evaluate the Computed Values

Calculate the values:- \( g(-1.1) = (-1.1 - 1)((-1.1)^2 + 1) = (-2.1)(2.21) = 4.641 \)- \( g(-1.01) = (-1.01 - 1)((-1.01)^2 + 1) = (-2.01)(2.0201) = 4.0605 \)- \( g(-1.001) = (-1.001 - 1)((-1.001)^2 + 1) = (-2.001)(2.002001) \approx 4.004 \)These calculations show that as \( x \to -1^- \), \( g(x) \) seems to approach 4.

Step 5: Conclude the Limit

Based on the values calculated, as \( x \) approaches \(-1\) from the left side, the values of \( g(x) \) are consistently getting closer to 4. This suggests that the limit of \( g(x) \) as \( x \to -1^- \) is 4.

Key Concepts

Factoring Polynomials

To solve problems involving the limits of functions like the one given, it's often useful to start by factoring any polynomial expressions. Factoring is the process of breaking down a polynomial into simpler components (called factors) that can be multiplied together to get the original polynomial.
For example, in our original function \[g(x) = \frac{x^4 - 1}{x + 1}\]the numerator can be factored. The expression \(x^4 - 1\) is a difference of squares, which can initially factor into:

• \((x^2 - 1)(x^2 + 1)\)

Then, recognize that \(x^2 - 1\) is also a difference of squares:

• \((x - 1)(x + 1)\)

Thus, the complete factorization of \(x^4 - 1\) is

• \((x - 1)(x + 1)(x^2 + 1)\).

Factoring is crucial because it allows us to simplify the expression and potentially eliminate terms that cause the original function to be undefined.

Function Simplification

After factoring, the next step in finding the limit of a function is simplifying the expression by canceling terms. In the provided problem, we have:\[g(x) = \frac{(x - 1)(x + 1)(x^2 + 1)}{x + 1}\]Here, you can cancel out the \((x + 1)\) term that appears in both the numerator and the denominator. This leaves us with:

• \(g(x) = (x - 1)(x^2 + 1)\) for \(x eq -1\)

It's important to note that this simplification is valid only for \(x eq -1\) because originally the function was undefined at \(x = -1\). Simplifying expressions helps us in analyzing functions' behavior more easily when approaching limits.

Table of Values

Creating a table of values can aid in numerical analysis of how a function behaves as it approaches a certain point. In this exercise, we want to evaluate:

• \(g(x) = (x - 1)(x^2 + 1)\)

as \(x\) approaches \(-1\) from the left (§ -1^-) by testing values close to \(-1\). Imagine selecting points such as \(-1.1, -1.01, -1.001\), and substituting them into the simplified function. You would calculate values like:

• \(g(-1.1) = 4.641\)
• \(g(-1.01) = 4.0605\)
• \(g(-1.001) \approx 4.004\)

The closer \(x\) gets to \(-1\), the closer \(g(x)\) approaches a single number. This approach helps confirm the behavior indicated by the limit.

Approaching Limits

The concept of limits is crucial in calculus, used to describe the behavior of a function as it approaches a specific point. For the given function, we looked at the limit as \(x\) approached \(-1\) from the left. Through calculating values such as \(g(-1.1), g(-1.01),\) and \(g(-1.001)\), we observed \(g(x)\) getting closer to 4.
The limit notation for this behavior is written as:

• \(\lim_{x \to -1^-} g(x) = 4\)

This means, even though \(g(x)\) is undefined exactly at \(x = -1\), the values of the function get arbitrarily close to 4 as \(x\) approaches \(-1\) from the left. Understanding limits helps in analyzing function continuity and detecting possible points of discontinuity.