Problem 6 - Thomas Calculus

Question

$$Let g(x)=\sqrt{x} \sin (1 / x)$$ $$\begin{array}{l}{\text { a. Does } \lim _{x \rightarrow 0^{+}} g(x) \text { exist? If so, what is it? If not, why not? }} \\ {\text { b. Does } \lim _{x \rightarrow 0^{+}} g(x) \text { exist? If so, what is it? If not, why not? }} \\\ {\text { c. Does } \lim _{x \rightarrow 0} g(x) \text { exist? If so, what is it? If not, why not? }}\end{array}$$

Step-by-Step Solution

Answer

a. Yes, \( \lim_{x \rightarrow 0^+} g(x) = 0 \). b. Yes, \( \lim_{x \rightarrow 0^+} g(x) = 0 \). c. No, the limit does not exist.

Step 1: Analyze the Function g(x)

The function given is \( g(x) = \sqrt{x} \sin(\frac{1}{x}) \). For \( x \to 0^+ \), \( \sqrt{x} \to 0 \) and \( \sin(\frac{1}{x}) \) oscillates between -1 and 1.

Step 2: Evaluate the Limit from the Right (Part a and b)

As \( x \rightarrow 0^+ \), the term \( \sqrt{x} \rightarrow 0 \) acts as a damping factor. Hence, the product \( g(x) = \sqrt{x} \sin(\frac{1}{x}) \to 0\), because the amplitude of \(\sin(\frac{1}{x})\) is bounded and approaches zero as \(\sqrt{x}\) approaches zero. Therefore, \( \lim_{x \rightarrow 0^+} g(x) = 0 \).

Step 3: Consider the Limit from Both Directions (Part c)

Given the result from the right-hand limit, check if approaching from the left differs. Since \( g(x) \) is only defined for \( x > 0 \), the limit from the left does not exist. Thus, the overall limit at \( x = 0 \) does not exist due to the absence of a path from \( x < 0 \).

Key Concepts

Oscillating Functions

Oscillating functions are fascinating because their values constantly change, behaving like a pendulum swinging back and forth. The sine function, written as \( \sin(x) \), is a common oscillating function in calculus. It continuously moves between -1 and 1. This oscillation can make finding limits challenging, especially as \( x \) approaches certain values.In the function \( g(x) = \sqrt{x} \sin(1/x) \), the term \( \sin(1/x) \) oscillates rapidly as \( x \) approaches zero from the right. Because of its nature, \( \sin(1/x) \) does not settle at a single value, making it misleadingly erratic at what seems to be a critical point.When you multiply \( \sin(1/x) \) by \( \sqrt{x} \), however, the oscillation seems less wild due to the damping effect of \( \sqrt{x} \). The result is that the product stabilizes somewhat as \( x \rightarrow 0^+ \). This damping by\( \sqrt{x} \), a term decreasing towards zero, effectively reduces the swings of \( \sin(1/x) \). Thus, an oscillating component in a function complicates limit evaluations but often interacts interestingly with other components to yield meaningful results.

One-Sided Limits

One-sided limits are useful when you need to understand function behavior from a particular direction. These limits involve approaching a point along the real number line either from the left or the right. They provide a way to consider behavior at boundaries or discontinuities.In calculus, you often distinguish between a limit from the right side, noted as \( \lim_{x \rightarrow a^+}f(x) \), and a limit from the left side, denoted as \( \lim_{x \rightarrow a^-}f(x) \).For the function \( g(x) = \sqrt{x} \sin(1/x) \), examining one-sided limits around zero is crucial. Since \( g(x) \) isn't defined for negative values of \( x \), its behavior as \( x \rightarrow 0^- \) is irrelevant; there is no path through \( x < 0 \) for \( g(x) \).Thus, only the right-side limit is significant. As we approach zero from the right, \( \lim_{x \rightarrow 0^+} \sqrt{x} \sin(1/x) \) simplifies due to \( \sqrt{x} \) going to zero. Despite the oscillations from \( \sin(1/x) \), the diminishing magnitude of \( \sqrt{x} \) allows the product's limit to be zero.

Limits of Composite Functions

Composite functions are compositions of two or more functions, where the output of one function becomes the input of another. Understanding limits of these functions involves analyzing each component's behavior and how they interact.In the given exercise, the function \( g(x) = \sqrt{x} \sin(1/x) \) is a composite of the square root function and the sine function, evaluated at \( 1/x \).To find its limit, you need to look at each part separately:

• \( \sin(1/x) \) is an oscillating function with values peaking between -1 and 1.
• \( \sqrt{x} \) trends towards zero as \( x \) approaches zero from the positive side.

The interaction between these components creates a scenario where the oscillating component is dampened. Despite \( \sin(1/x) \) oscillating, the overall product's limit depends more on \( \sqrt{x} \) because it moves towards zero quickly.For composite functions, focus on how the outer function reshapes the values of the inner function. In this instance, the limit \( \lim_{x \rightarrow 0^+} g(x) = 0 \) results from \( \sqrt{x} \) fully counteracting the oscillations. This type of damping and interaction between functions is the essence of limit evaluation in composite functions.