Problem 24 - 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 \(-\infty\). $$ \lim _{x \rightarrow-\infty}\left(2+\frac{1}{x^{2}}\right) $$

Step-by-Step Solution

Answer

\( \lim_{x \rightarrow -\infty} \left( 2 + \frac{1}{x^2} \right) = 2 \)

Step 1: Understand the Function

The function provided is: \[ f(x) = 2 + \frac{1}{x^2} \]

Step 2: Analyze the Behavior as x Approaches -\ ( \text{infinity} )

As \( x \rightarrow - \infty \), the term \( \frac{1}{x^2} \) approaches zero because \( x^2 \) becomes very large, making the fraction very small. Since the denominator is squared, it is always positive and thus approaches zero from the positive side.

Step 3: Combine Results in the Context

As \( x \rightarrow - \infty \), the term \( \frac{1}{x^2} \) approaches zero. Therefore, the function \( f(x) = 2 + \frac{1}{x^2} \) approaches \( 2 + 0 = 2 \).

Step 4: Conclude the Limit

Based on the analysis, \[ \lim _{x \rightarrow -\infty} \left(2 + \frac{1}{x^2}\right) = 2 \]

Key Concepts

limits

In calculus, a limit describes the value that a function approaches as the input (or variable) approaches a particular point. Limits are essential for defining derivatives and integrals. To find a limit, we often look at the behavior of the function as the variable approaches a specific value.

For example, in our given problem, we need to find the limit of \(2 + \frac{1}{x^2}\) as \(x\) approaches \(-\infty\). This involves understanding how each term in the function behaves when \(x\) gets very large in the negative direction. By analyzing each term separately, we combine the results to determine the overall limit.

Calculating limits can sometimes involve more complex techniques like L'Hopital's rule, which is used when limits result in indeterminate forms like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). However, in the case of our exercise, the limit is straightforward to find by basic analysis of terms.

infinity

Infinity is not a number but a concept describing something without any bound or end. In mathematics, it is used to express the idea of endlessness. There are two types of infinity: positive infinity \(+\infty\) and negative infinity \(-\infty\).

Positive infinity refers to values increasing without bound, whereas negative infinity refers to values decreasing without bound.

In the given exercise, we consider the limit of the function as \(x\) approaches \(-\infty\). When dealing with such limits, it's essential to observe how the terms in the function behave as \(x\) becomes very large (in the positive or negative direction). For instance, as \(x\) becomes very large negatively, the term \(\frac{1}{x^2}\) approaches 0 because \(x^2\) increases very quickly. Understanding these behaviors helps us determine the overall limit.

behavior of functions

To find the limit of a function, we need to analyze the behavior of the function as the variable approaches a specific value or infinity. The behavior of functions at infinity tells us how the function acts when the variable becomes very large in positive or negative directions.

In the example \(2 + \frac{1}{x^2}\), as \(x\) approaches \(-\infty\), we focus on the term \(\frac{1}{x^2}\).

- When \(x\) becomes very large negatively (approaching \(-\infty\)), \(\frac{1}{x^2}\) approaches 0 because the denominator becomes extremely large.
- Since \(\frac{1}{x^2}\) approaches 0, the entire function \(2 + \frac{1}{x^2}\) approaches \(2 + 0 = 2\).

This behavior analysis is crucial in determining limits. Knowing how each term behaves as \(x\) gets very large or very small helps us understand the function's behavior at those points. This approach is fundamental in calculus for finding limits and understanding the overall function's behavior.