Problem 23 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the indicated one-sided limit, if it exists. \(\lim _{x \rightarrow 2^{-}} \frac{x-3}{x+2}\)

Step-by-Step Solution

Answer

The one-sided limit of the function \(f(x) = \frac{x-3}{x+2}\) as x approaches 2 from the left is \(-\frac{1}{4}\).

Step 1: Identify the one-sided limit

We are given the limit: \(\lim _{x \rightarrow 2^{-}} \frac{x-3}{x+2}\) This is asking for the behavior of the function as x approaches 2 from the left (smaller values of x).

Step 2: Substitute the value of x and check if the function is undetermined

First, we have to substitute x by 2: \(\frac{2-3}{2+2} = \frac{-1}{4}\) We realize that the function is not undefined and thus we can solve the one-sided limit directly.

Step 3: Calculate the limit

Now, we just found that the limit of the function as x approaches 2 from the left is \(-\frac{1}{4}\): \(\lim _{x \rightarrow 2^{-}} \frac{x-3}{x+2} = -\frac{1}{4}\) Therefore, the one-sided limit of the function \(f(x) = \frac{x-3}{x+2}\) as x approaches 2 from the left is \(-\frac{1}{4}\).

Key Concepts

One-Sided Limits

One-sided limits help us understand how a function behaves as the input (x) approaches a certain value from one specific side, either from the left or the right. In this exercise, we're dealing with a left-sided limit, indicated by the notation \( \lim_{x \rightarrow 2^{-}} \). This means we're interested in the behavior of the function as x approaches 2 from values less than 2.

Understanding one-sided limits can be crucial because sometimes, the behavior of a function can be different when approached from different directions. A function may not have a two-sided limit but can still have one-sided limits on either side. For practical purposes:

• Use the limit notation with a minus sign \((-)\) for approaching from the left.
• Use a plus sign \((+)\) for approaching from the right.

Evaluating one-sided limits can often reveal discontinuities or vertical asymptotes present in a function, making these calculations important in understanding function continuity and behavior.

Limit Evaluation

Limit evaluation is the process of finding the limit of a function as it approaches a specific point. This exercise involves evaluating the one-sided limit of the function \( \frac{x-3}{x+2} \) as x approaches 2 from the left. The primary step in limit evaluation is to substitute the value into the function, if possible.

Upon substitution, it is crucial to check if the function becomes undefined or results in an indeterminate form like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). If the function is defined, such as in this exercise where substituting 2 gives \( \frac{-1}{4} \), the limit can be evaluated directly.

In cases where direct substitution leads to an undefined value, alternative methods might be needed:

• Factorization to simplify the function.
• Rationalizing to remove square roots.
• Applying L'Hôpital's Rule for indeterminate forms.

For this exercise, since no complexity arose from substitution, we quickly reached the evaluated limit, \( -\frac{1}{4} \).

Function Behavior

Understanding function behavior near a point helps us predict how the function will act when approaching that value. This is particularly useful in calculus, where examining these behaviors assists with defining concepts like continuity and differentiability.

In this exercise, examining the function \( \frac{x-3}{x+2} \) as x approaches 2 from the left gives a value of \( -\frac{1}{4} \). This tells us the function approaches a specific value smoothly and without abrupt changes. Analyzing the numerators and denominators of rational functions often gives insights into:

• The tendency for vertical asymptotes if the denominator approaches zero while the numerator does not.
• Horizontal asymptotes signaling the end behavior as x moves towards \( \infty \) or \( -\infty \).

In summary, by studying the limit and function behavior, we gain an understanding of how the function marches towards a particular number and behaves around it, helping detect discontinuities and asymptotic behavior.