Problem 48 - College Algebra and Calculus: An Applied Approach

Question

find the limit $$ \lim _{x \rightarrow-1} \frac{x^{3}-1}{x+1} $$

Step-by-Step Solution

Answer

The limit as x approaches -1 is 1.

Step 1: Factorize the Numerator

The numerator of the function can be factorized: \(x^3 - 1\) can be rewritten as \((x-1)(x^2 + x + 1)\). The expression becomes: \(\frac{(x-1)(x^2 + x + 1)}{x+1}\)

Step 2: Simplify the Expression

Notice that (x - 1) and (x + 1) are both factors, so they cancel each other out. This leaves us with: \(x^2 + x + 1\)

Step 3: Evaluate the Limit

Plug in -1 for x in the expression \(x^2 + x + 1\). This gives you \((-1)^2 -1 + 1 = 1\).

Key Concepts

Limits and Continuity

In calculus, understanding the concept of limits is fundamental. A limit captures the behavior of a function as the input approaches a certain value. Continuity, on the other hand, means that as you trace along the graph of the function, you can do so without lifting your pencil, indicating there are no breaks or holes.

When it comes to finding limits, if substituting the value directly into the function does not work due to an indeterminate form like 0/0, we then explore other methods such as factoring, which leads to simplifying the expression to evaluate the limit.

For example, in the exercise \( \lim _{x \rightarrow -1} \frac{x^3 - 1}{x + 1} \), the direct substitution is not possible because it leads to a 0/0 scenario. Therefore, we must perform additional steps to figure out the behavior of the function as x approaches -1, which is what continuity and limit evaluation are all about.

Factoring Polynomials

Factoring polynomials is a critical skill for manipulating algebraic expressions, especially when evaluating limits in calculus. To factor a polynomial is to rewrite it as a product of simpler polynomials whose values agree with the original polynomial for all inputs.

The process usually involves identifying patterns or using techniques like the difference of cubes, which is applied in our exercise. The cubic term \( x^3 - 1 \) is recognized as a difference of cubes because it can be written as \( x^3 - 1^3 \) and factored into \( (x - 1)(x^2 + x + 1) \).

This step is crucial because it allows us to simplify the expression, making it easier to find the limit as x approaches the given value.

Simplifying Expressions

After factoring polynomials, the next step in evaluating limits is to simplify the expression. Simplification might entail canceling out common factors between the numerator and the denominator, reducing fractions, or performing arithmetic operations to minimize the expression complexity.

In our example, once the polynomial is factored, we identify that \( x+1 \) in the denominator cancels out one of the factors in the numerator, simplifying the expression to \( x^2 + x + 1 \). Simplifying the expression is a vital process as it often leads to an easier evaluation of the limit and helps in better understanding the behavior of the function near the point of interest.

Evaluating Limits

Evaluating limits is the act of determining the value a function approaches as the input gets infinitely close to a certain point. This concept is not always straightforward, since directly substituting the value into the function does not always work.

Once an expression is simplified, like in our problem, evaluating the limit becomes straightforward. You substitute \( x = -1 \) into the simplified expression \( x^2 + x + 1 \) to find the limit as \( 1 \). This evaluation confirms the behavior of the function at the point \( x = -1 \) and, provided the original expression was continuous at this point, would also be the function's value there.