Question

Plot the functions \(u(x), l(x)\), and \(f(x) .\) Then use these graphs along with the Squeeze Theorem to determine \(\lim _{x \rightarrow 0} f(x)\). $$ u(x)=|x|, l(x)=-|x|, f(x)=x \sin (1 / x) $$

Short answer

The limit is 0 as \( x \rightarrow 0 \).

Step-by-step solution

Understanding the Functions

Given the functions are: \( u(x) = |x| \), which creates a V-shaped graph with the point at the origin, and \( l(x) = -|x| \), which is an inverted V. The function \( f(x) = x \sin(1/x) \) oscillates rapidly as \( x \) approaches zero due to the \( \sin(1/x) \) term.

Plotting the Functions

Plot these functions on a graph. - \( u(x) = |x| \) is a V-shaped graph with points (0,0) and symmetric about the y-axis. - \( l(x) = -|x| \) is the reflection of \( u(x) \) about the x-axis. - \( f(x) = x \sin(1/x) \) will oscillate between \( l(x) \) and \( u(x) \).

Applying the Squeeze Theorem

We know that \( -|x| \leq x \sin(1/x) \leq |x| \) for all \( x eq 0 \). As \( x \rightarrow 0 \), both \( u(x) \rightarrow 0 \) and \( l(x) \rightarrow 0 \), which means, by the squeeze theorem, \( f(x) \rightarrow 0 \).

Finding the Limit

By applying the Squeeze Theorem, since \( u(x) \) and \( l(x) \) both approach 0, \( \lim_{x \to 0} f(x) = 0 \) because \( f(x) \) is squeezed between \( u(x) \) and \( l(x) \).

Key concepts

Functions

In calculus, functions represent relationships between sets of numbers or variables. They assign each input exactly one output. Functions come in many forms, like linear, quadratic, or oscillatory forms, each exhibiting unique behaviors. In the context of the given problem:

• The function \(u(x) = |x|\) represents the absolute value function. It generates a V-shaped graph, symmetrical around the y-axis, with a pointed vertex at the origin (0,0).
• The function \(l(x) = -|x|\) is the negative of the absolute value function, flipping the V-shape upside down, forming an inverted V.
• The function \(f(x) = x \sin(1/x)\) combines a linear factor with an oscillatory sine function. This results in rapid oscillations as \(x\) approaches zero, creating a complex behavior.

Understanding each function's behavior is crucial for plotting and analyzing limits using the Squeeze Theorem.

Limit

The concept of a limit is central to calculus, helping to describe the behavior of a function as it approaches a particular point. In this exercise, we are interested in finding the limit of \(f(x) = x \sin(1/x)\) as \(x\) approaches 0.
A limit expresses the value that a function approaches as its input approaches a certain point. When approaching zero, the complications arise due to oscillations in \( \sin(1/x)\), but the limit can still be determined using strategies like the Squeeze Theorem.
In simple terms, the Squeeze Theorem allows us to "squeeze" the function of interest between two simpler functions whose limits are easier to find. Since both \(u(x)\) and \(l(x)\) approach 0 as \(x\) approaches 0, they constrain the oscillating \(f(x)\), leading to the conclusion that \(\lim_{x \to 0} f(x) = 0\).

Graph Plotting

Graph plotting is a visual method to understand and analyze equations and their behaviors. It involves creating a visual representation of the various functions.
1. **Plotting \(u(x) = |x|\)**: This graph is V-shaped, symmetric about the y-axis with the tip at the origin (0,0). It demonstrates that as \(x\) moves away from zero, the function value increases linearly.
2. **Plotting \(l(x) = -|x|\)**: This function mirrors \(u(x)\) across the x-axis. It also starts from the origin and decreases linearly as \(x\) moves away from zero, showing the same form but inverted.
3. **Plotting \(f(x) = x \sin(1/x)\)**: This graph shows rapid oscillations which become tighter as \(x\) approaches zero. It reflects the fluctuating nature due to the sine component.
Plotting these graphs helps visually verify the behavior of the functions, illustrating how \(f(x)\) is trapped or "squeezed" between \(u(x)\) and \(l(x)\). This is critical in applying the Squeeze Theorem effectively.

Oscillation

Oscillation refers to the repetitive variation, typically in time, of some measure about a central value or between two or more different states. It is a key feature in the function \(f(x) = x \sin(1/x)\).
This function manifests oscillation due to the \(\sin(1/x)\) component, which oscillates indefinitely as \(x\) approaches zero. The period of oscillation diminishes, creating more rapid fluctuations close to zero.

• The \(\sin(1/x)\) term causes these oscillations because its argument \(1/x\) grows large as \(x\) nears zero, creating very fast oscillations.
• The factor \(x\) outside the sine function scales these oscillations, so they shrink in amplitude as \(x\) decreases.

Understanding oscillation is essential to determining the behavior of \(f(x)\) and employing the Squeeze Theorem, as it helps justify why \(f(x)\) doesn't diverge and is bounded by \(u(x)\) and \(l(x)\).