Problem 50 - Calculus Early Transcendentals: Pearson New International Edition

Question

Many software packages have programs for calculating limits, although you should be warned that they are not infallible. To develop confidence in your program, use it to recalculate some of the limits in Problems 1-28. Then for each of the following, find the limit or state that it does not exist. $$ \lim _{x \rightarrow 0}|x|^{x} $$

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Express the Limit in Exponential Form

The expression inside the limit is \(|x|^x\). We can rewrite this as an exponential function for easier differentiation and analysis. Using logarithms, we have \(|x|^x = e^{x \ln |x|}\). This transforms our limit into: \[ \lim_{x \to 0} |x|^x = \lim_{x \to 0} e^{x \ln |x|}. \] Our task now is to find the limit of the exponent.

Step 2: Analyze the Exponent

Consider the expression \(x \ln |x|\) as \(x\) approaches 0. Note that \(|x|\) is symmetrical around 0, meaning it behaves as \[ x \ln |x| = x \ln x \text{ for } x > 0 \text{ and } x \ln (-x) \text{ for } x < 0. \] We examine \(x \ln x\) first as \(x \) approaches 0 from the positive side by letting \(x = e^{-t}\). As \(x \rightarrow 0^{+}\), \(t \rightarrow \infty\), and \(x \ln x = \frac{-t}{e^t}\) goes to 0.

Step 3: Consider the Limit Symmetrically Around Zero

Since \(|x| = x\) for \(x > 0\) and \(|x| = -x\) for \(x < 0\), the function \(x \ln |x| \) for negative values behaves similarly to when it is positive due to the even nature of the logarithm around negative values. Hence, the expression \(x \ln |x|\) tends to 0 in both positive and negative approaches towards 0, i.e., \[ \lim_{x \to 0} x \ln |x| = 0. \]

Step 4: Apply the Exponential Function Limit

Since we found that \(\lim_{x \to 0} x \ln |x| = 0\), we use the continuity of the exponential function to establish that \[ \lim_{x \to 0} e^{x \ln |x|} = e^0 = 1. \] Thus, the original limit becomes \(1\) as well.

Key Concepts

Limits

In calculus, the concept of limits is fundamental for understanding how functions behave as their inputs approach certain values. Limits help us describe and define discontinuities, behavior at infinity, and nearby values that are difficult to calculate directly.

You might wonder why the limit of a function is significant: it's a powerful tool that lets us infer about a function's behavior at points where it isn't explicitly defined. The limit of a function as it approaches a particular point is like peeking at the behavior just around the pinpoint area, providing insight into the immediate trend.

Limits can be classified mainly into two types:

• Finite limits: Analysis close to known points within the function's domain.
• Infinite limits: Exploration as inputs extend towards infinity or negative infinity.

If a limit exists, it means the function approaches a specific value as the input nears a particular point. When the input tends to zero, such as in the exercise, finding the limit can indicate whether the function's output stabilizes, shoots off to infinity, or behaves unpredictably.

Exponential Functions

Exponential functions are a type of mathematical function where a constant base is raised to a variable exponent. One of the most popular bases in calculus is the natural number, denoted as \( e \), approximately equal to 2.718.

Exponential functions are significant in various fields, like biology, finance, and physics, due to their ability to model growth or decay processes succinctly and accurately. These functions show exponential growth if the exponent increases or exponential decay if the exponent decreases.

In the context of limits, exponential functions show interesting behavior, essential when calculating limits of certain forms. For example, in the exercise, \(|x|^x\) is rewritten using exponential notation as \(e^{x \ln |x|}\). This transformation makes the limit more analyzable since the exponential function, \( e^x \), is continuous and differentiable everywhere, ensuring smoother analysis.

Understanding exponential functions includes knowing their unique properties:

• They grow rapidly as the variable increases.
• They never reach zero, but get infinitely close (particularly true for decay functions).
• The base \( e \) has nice calculus properties: its derivative and integral relate closely with each other.

These distinctive features of exponential functions simplify many aspects of calculus, particularly when working with growth models and limits similar to the one in the exercise.

Logarithms

Logarithms are the inverse operations of exponentiation. In simple terms, if you have an equation in exponential form like \(b^y = x\), its logarithmic form will be \( \log_b(x) = y \). This transformation allows for solving equations where the unknown is an exponent.

In calculus, logarithms, especially natural logarithms \( \ln \), are incredibly valuable for manipulating and simplifying complex expressions. They make the impossible simplify and the abstract to become calculable in many calculus problems.

In the exercise, transforming \(|x|^x\) into \(e^{x \ln |x|}\) uses logarithms to easily differentiate and analyze the expression. The natural logarithm is especially helpful because it's particularly integrable and differentiable, making it efficient in finding the derivatives of more complex calculus functions.

Key properties of logarithms include:

• Convert multiplicative processes into additive ones: \(\log_b(mn) = \log_b(m) + \log_b(n)\).
• Inverse nature with exponentials: \(e^{\ln(x)} = x\).
• Simplifying exponents: \(\log_b(x^r) = r \cdot \log_b(x)\).

These attributes of logarithms offer powerful methods to break down complicated functions, allowing mathematicians and students alike to delve deeper into the theoretical confines of a problem with increased precision and confidence.