Problem 33 - College Algebra and Calculus: An Applied Approach

Question

Find the limit. $$ \lim _{x \rightarrow-2} \frac{x^{2}-1}{2 x} $$

Step-by-Step Solution

Answer

The limit is \( \frac{3}{4} \)

Step 1: Factorize the numerator

The numerator \( x^{2}-1 \) can be factored as \( (x-1)(x+1) \). So rewrite the function. \( \lim _{x \rightarrow-2} \frac{(x-1)(x+1)}{2x} \)

Step 2: Substitute x = -2

If a function is defined at a point, you can find its limit at this point by simply substituting this value into the function. Now, substitute \( x = -2 \) into the function. \( \lim _{x \rightarrow-2} \frac{(-2-1)(-2+1)}{2*-2} = \frac{-3*(-1)}{-4} = \frac{3}{4} \)

Step 3: Write down the limit result

The limit of the function \( \frac{x^{2}-1}{2 x} \) as \( x \) approaches -2 is \( \frac{3}{4} \).

Key Concepts

Factorization

In the realm of calculus, factorization is a handy tool when working with polynomial expressions, especially when it comes to finding limits. This method involves breaking down a complex expression into simpler components or factors. In our exercise, the polynomial in the numerator \( x^2 - 1 \) is factorized into \( (x-1)(x+1) \).
Factorization is crucial when you're dealing with expressions that are not straightforward to evaluate directly. By simplifying the expression, we make it easier to handle or even eliminate indeterminate forms leading to clearer results.

• It helps simplify the polynomial expressions.
• Reveals hidden common factors to simplify expressions further.

In cases like ours, where direct substitution can result in undefined expressions, factorization is often the first step in simplifying the limit process.

Substitution Method

The substitution method is a fundamental concept in evaluating limits and serves as a direct way of finding the limit of a function as the variable approaches a specific value. Once an expression is simplified using methods like factorization, substituting the point of interest often allows us to evaluate the limit directly. This is exactly what we do in our example.
Upon simplifying the expression to \( \frac{(x-1)(x+1)}{2x} \), we can substitute \( x = -2 \) into the expression. This substitution is straightforward:

• Replace \( x \) with \( -2 \) giving \( \frac{(-2-1)(-2+1)}{2(-2)} \).
• Calculate step by step to ensure clarity and accuracy.

Substitution checks for continuity of the function at a given point. If it's continuous, then substitution is all you need to determine limits.

Evaluating Limits

Evaluating limits helps us understand the behavior of a function as the input value approaches a certain point. This process tells us how the function behaves nearby, even if it doesn’t exist exactly at that point. In our exercise, after simplifying and using substitution, we find that the limit is \( \frac{3}{4} \).
Evaluating limits:

• Provides insight into the local behavior of functions.
• Vital for determining points of continuity or discontinuity.

Remember, in some cases, you might encounter indeterminate forms, which is where simplifying or using l'Hôpital's rule becomes essential. Always simplify expressions wherever possible to find accurate limits.