Problem 27 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the indicated one-sided limit, if it exists. \(\lim _{x \rightarrow 0^{+}} \frac{x-1}{x^{2}+1}\)

Step-by-Step Solution

Answer

The right-sided limit of the function \(f(x) = \frac{x-1}{x^2+1}\) as x approaches 0 is -1. Therefore, \(\lim _{x \rightarrow 0^{+}} \frac{x-1}{x^{2}+1} = -1\).

Step 1: Identify the function and the limit

In this problem, we need to find the right-sided limit of the function \(f(x) = \frac{x-1}{x^2+1}\) as x approaches 0. This is written as: \(\lim _{x \rightarrow 0^{+}} \frac{x-1}{x^{2}+1}\)

Step 2: Analyze the behavior of the function as x approaches 0 from the right

As x approaches 0 from the right, we can replace x with values slightly greater than 0 and see how the function behaves. For example, we could try \(x=0.1, x=0.01, x=0.001\) and so on. Observe that: \(f(0.1) \approx -0.989\) \(f(0.01) \approx -0.9998\) \(f(0.001)\approx -0.999998\) By analyzing the behavior of the function as x gets closer to 0 from the right, it's clear that the function is approaching -1.

Step 3: State the limit

Based on our observation, the right-sided limit of the function as x approaches 0 is -1. Therefore, we can write the limit as: \(\lim _{x \rightarrow 0^{+}} \frac{x-1}{x^{2}+1} = -1\)

Key Concepts

Limit of a Function

Understanding the limit of a function is crucial in calculus because it helps to comprehend the behavior of functions as they approach a particular point. The limit of a function at a certain point is essentially the value that the function approaches as the input (or the x-value) gets closer to that point. It's important to note that the limit doesn't always equal the function's value at that point; in fact, the function doesn't even need to be defined at the limit point.

When solving \( \lim _{x \rightarrow a} f(x) \), we are not so much interested in what happens at \( x=a \) but rather the value that \( f(x) \) is getting closer and closer to as \( x \) approaches \( a \) from both sides.

Right-Sided Limit

A right-sided limit, denoted as \( \lim _{x \rightarrow a^{+}} f(x) \), is the value a function approaches as the input approaches a certain number, \( a \), from the right (that is, from values greater than \( a \) ).

To find a right-sided limit like \( \lim _{x \rightarrow 0^{+}} \frac{x-1}{x^{2}+1} \), we consider only the values of \( x \) that are just slightly larger than 0. This particular focus allows us to observe the trend or pattern the function takes as we get closer to, but never actually reach, the point where \( x=0 \) from the positive side. As seen in the exercise, this process enables us to deduce that the function approaches -1 as we come infinitesimally close to 0 from the right side.

Limit Approaching Behavior

The approaching behavior of a function as \( x \) nears a particular value can reveal a wealth of information about the function's characteristics in a neighborhood around that point. By evaluating the limit approaching behavior, such as \( \lim _{x \rightarrow a} f(x) \), we are analyzing how \( f(x) \) behaves as \( x \) draws nearer to \( a \) without actually being \( a \) itself. This is particularly useful in cases where \( f(x) \) might not be defined at \( x=a \) or the function exhibits a discontinuity at that point.

The patterns we discern through this investigation, such as in our exercise with values like 0.1, 0.01, and 0.001, can be quite telling. They suggest a trend towards a specific value which we identify as the limit. The consistency in the function's behavior as it approaches the point from one side gives us confidence in stating the one-sided limit's value.

Calculus for the Managerial Life and Social Sciences

Calculus is not just a subject for engineers and physicists; it has practical applications in fields such as business, economics, and the social sciences. When dealing with 'calculus for the managerial life and social sciences,' concepts like limits prove essential for understanding rates of change, optimization problems, and behaviors of cost and profit functions near critical points.

In the context of managerial life, comprehending right-sided limits could play a role in predicting the immediate effects of pricing strategies as demand approaches a threshold. Social scientists might use limits to analyze the immediate impact of policy changes on population behaviors. The exercise example showcases how limits apply in a variety of contexts, assisting decision-makers in forecasting trends based on mathematical models.