Problem 40 - Calculus

Question

What does a calculator suggest about \(\lim _{x \rightarrow 0^{+}} x e^{1 / x} ?\) Does the limit appear to exist? Explain.

Step-by-Step Solution

Answer

The limit does not appear to exist as a finite number; it tends to infinity.

Step 1: Understanding the Function

We have the function \( f(x) = x e^{1/x} \). We want to find the limit of this function as \( x \) approaches 0 from the positive side, i.e., \( \lim _{x \rightarrow 0^{+}} x e^{1 / x} \).

Step 2: The Behavior of \( e^{1/x} \)

Consider the term \( e^{1/x} \) as \( x \rightarrow 0^{+} \). As \( x \) gets closer to 0, \( 1/x \) becomes very large. Consequently, \( e^{1/x} \) becomes extremely large very quickly.

Step 3: Multiplying by \( x \)

Next, consider \( x \cdot e^{1/x} \). As \( x \rightarrow 0^{+} \), \( x \) itself approaches 0. Despite \( e^{1/x} \) increasing towards infinity, \( x \) tries to pull the whole product towards zero.

Step 4: Evaluating the Product

For very small values of \( x \), the value of \( x \cdot e^{1/x} \) depends on which portion of the expression grows or shrinks faster. Since \( e^{1/x} \) grows significantly faster than \( x \) can shrink, the product \( x \cdot e^{1/x} \) grows without bound towards infinity.

Step 5: Limit Conclusion

Given that \( x e^{1/x} \) grows without bound as \( x \rightarrow 0^{+} \), we can conclude that \( \lim _{x \rightarrow 0^{+}} x e^{1 / x} = \infty \).

Key Concepts

Limits

In calculus, limits help us understand the behavior of functions as they approach specific points or infinity. When investigating a limit, we want to know what value a function gets closer to as the input approaches a particular number. Let's break down the essential ideas associated with limits:

• Approaching a Point: We often consider what happens as the input gets closer to a specific value, but not necessarily reaching it. For \(x e^{1/x}\), we are interested in what happens as \(x \) approaches \(0^{+}\).
• One-Sided Limits: This type of limit only considers the approach from one direction. Here, \(x \) is approaching 0 from the positive side, indicated by \(0^{+}\).
• Limit Behavior: Sometimes, functions stabilize to a finite number; other times, they might grow indefinitely large or small. In our problem, \(\lim_{x \to 0^{+}} x e^{1/x} = \infty\) means the function increases without bound as \(x \) gets closer to 0 from the positive side.

Limits are fundamental in defining derivatives, integrals, and understanding continuity. They're crucial in evaluating how variables interact at boundaries.

Exponential Functions

Exponential functions are a key concept in calculus and can be quite powerful. They involve the constant \(e\), approximately equal to 2.718, which forms the base of natural logarithms. These functions are expressed as \(e^x\). Here's what you need to know:

• Rapid Growth or Decay: The function \(e^x\) grows very rapidly as \(x\) increases. Conversely, \(e^{-x}\) exhibits fast decay as \(x\) rises.
• Utility in Calculus: Exponential functions are often used in solving differential equations and modeling exponential growth or decay processes in real-world situations such as population dynamics or radioactivity.
• Role in the Problem: In the expression \( e^{1/x}\), note that as \(x \) becomes smaller, \(1/x \) becomes larger, causing \(e^{1/x}\) to increase significantly. That's why this function accelerates the growth of \(x e^{1/x}\) as \(x \) approaches \(0^{+}\).

Understanding exponential functions aids in recognizing their behavior and potential impacts when combined with other functions.

Product of Functions

When combining functions through multiplication, the behavior of the resulting product depends on each function's contribution. This is particularly noteworthy when one function grows rapidly and the other shrinks. Let's explore this with our function \(f(x) = x e^{1/x}\):

• Interaction between Terms: Here, as \(x \) approaches 0 from the positive side, one part of the product, \(x\), trends towards zero. Meanwhile, \(e^{1/x}\) explodes towards infinity.
• Dominance in Growth: To understand the limit behavior, we observe which component governs the product's behavior. Since \(e^{1/x}\) grows much faster than \(x\) shrinks, it dominates, causing the entire product to ascend towards infinity.
• Assessing the Limit: Evaluating \(\lim_{x \to 0^{+}} x e^{1/x} = \infty\) involves seeing the fast-growing component outweigh any shrinking tendency of \(x\).

In calculus, analyzing products of functions provides insights on how various behaviors impact a function's overall trend near specific points or infinity.