Problem 31 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

Either find the given limit or show it does not exist. If the limit is infinite, indicate whether it is \(+\infty\) or \(-\infty\). $$ \lim _{x \rightarrow 0^{-}} x \sqrt{1-\frac{1}{x}} $$

Step-by-Step Solution

Answer

The limit is \( -\infty \).

Step 1 – Analyze the Inner Function

First, consider the expression inside the square root: \( 1 - \frac{1}{x} \). As \( x \rightarrow 0^− \), \( \frac{1}{x} \rightarrow -\infty \). Therefore, \(1 - \frac{1}{x} \rightarrow +\infty \).

Step 2 – Evaluate the Square Root

Since the expression inside the square root goes to \( +\infty \), the square root of it also approaches \( +\infty \). Thus, \[ \sqrt{1 - \frac{1}{x}} \rightarrow \sqrt{+\infty} = +\infty. \]

Step 3 – Combine the Results

Now consider the product \( x \sqrt{1 - \frac{1}{x}} \). As \( x \rightarrow 0^- \), \( x \) is negative and very small, and combining this with a term that goes to \( +\infty \) gives \[ x \sqrt{1 - \frac{1}{x}} \rightarrow 0^- \times +\infty. \]

Step 4 – Determine the Sign of the Result

In this product, \( 0^- \times +\infty \) results in \(-\infty\) because a negative small number multiplied by a very large positive number is a very large negative number. Therefore, the limit is \( -\infty \).

Key Concepts

Limits

Limits help us understand the behavior of functions as they approach a specific point. In calculus, a limit describes the value a function approaches as the input approaches a certain point. By evaluating limits, we can predict the behavior of functions at points where they might not be explicitly defined. When we write \(ormalsize{\text{lim}}_{x \rightarrow a} f(x)\rightarrow L\ormalsize{\), it means that as \(x\) gets closer to \(a\), the function \(f(x)\) approaches the value \(L\). Sometimes, limits can be straightforward, other times they might involve more complex functions that require manipulation to understand their behavior at the point of interest.

Infinite Limits

Infinite limits are limits where the function increases or decreases without bound as the input approaches a certain value. This is different from regular limits that approach a finite value. For example, when we say \(ormalsize{\text{lim}}_{x \rightarrow a} f(x) = +ormalsize{\text{\textinfty}}\ormalsize{\), it means that as \(x\) gets closer to \(a\), \(f(x)\) increases without bound. Factors like negative or positive input values can influence whether an infinite limit will be positive or negative infinity. This process often involves understanding the signs and behaviors of the components within the function, as shown in our solution.

Calculus

Calculus is the branch of mathematics that studies rates of change (differentiation) and accumulation of quantities (integration). It's essential for understanding and describing changes, often using the concept of limits. In the context of limits, calculus helps provide tools and techniques to handle complex functions and their behaviors. Understanding how functions behave near specific points and using limits to predict this behavior is a critical part of calculus. These tools allow us to solve problems involving instantaneous rates of change, areas under curves, and more, making it a foundational discipline for many fields in science and engineering.