Question

Plot the functions \(u(x), l(x)\), and \(f(x) .\) Then use these graphs along with the Squeeze Theorem to determine \(\lim _{x \rightarrow 0} f(x)\). $$ u(x)=|x|, l(x)=-|x|, f(x)=x \sin \left(1 / x^{2}\right) $$

Short answer

The limit is 0.

Step-by-step solution

Understand the Functions

We are given three functions: \( u(x) = |x| \), \( l(x) = -|x| \), and \( f(x) = x \sin\left(\frac{1}{x^2}\right) \). We need to analyze these functions around the point \( x = 0 \).

Graph the Functions

Plot the graphs of \( u(x) = |x| \), \( l(x) = -|x| \), and \( f(x) = x \sin\left(\frac{1}{x^2}\right) \). The function \( u(x) \) forms a V-shape upward, \( l(x) \) forms a V-shape downward, and \( f(x) \) oscillates between the limits of \( -|x| \) and \( |x| \).

Apply the Squeeze Theorem

The Squeeze Theorem states that if \( l(x) \leq f(x) \leq u(x) \) for all \( x \) near some point, and \( \lim_{x \to 0} l(x) = \lim_{x \to 0} u(x) = L \), then \( \lim_{x \to 0} f(x) = L \). Here, as \( x \to 0 \), both \( u(x) \) and \( l(x) \) tend to 0, i.e., \( \lim_{x \to 0} u(x) = \lim_{x \to 0} l(x) = 0 \).

Conclude the Limit of f(x)

Since \( f(x) \) is squeezed between \( -|x| \) and \( |x| \) as \( x \to 0 \), and both limits are 0, we conclude by the Squeeze Theorem that \( \lim_{x \to 0} f(x) = 0 \).

Key concepts

Calculus

Calculus is a branch of mathematics that deals with change, using concepts like derivatives and integrals. It provides tools for analyzing functions and understanding the behavior of their graphs.
Two major aspects of calculus are differentiation and integration:

• Differentiation: This involves finding the derivative of a function, which describes the rate of change. For instance, if we have a curve on a graph, the derivative at any point gives the slope of the tangent to the curve at that point.
• Integration: This is about finding the area under a curve, which can represent total accumulated change.

When dealing with limits in calculus, you're often looking at the behavior of a function as it approaches a particular point. This is especially useful for functions that have undefined values at certain points, which is a common problem when working with trigonometric functions like sine and cosine.

Limits

Limits are a fundamental concept in calculus, providing a way to understand what happens to functions as inputs approach certain values. In our example with the squeeze theorem, we're interested in what happens to the function as it gets close to zero.
The limit of a function can help us determine the value a function gets close to, even if it never actually reaches that point. When evaluating:

• Consider any value the function might approach as the variable nears a specific point.
• Use strategies like graphing to visualize behavior near points of interest.
• If a function is bounded by two other functions that have the same limit at a certain point, we can conclude that our function also approaches that limit, which is precisely what the Squeeze Theorem tells us.

Mastering limits allows you to tackle complex calculus problems and aids greatly in predicting behavior of physical phenomena modeled by functions.

Trigonometric Functions

Trigonometric functions, including sine, cosine, and tangent, play a crucial role in calculus, particularly when analyzing oscillatory behavior in graphs. In the given exercise, the function \( f(x) = x \sin\left(\frac{1}{x^2}\right) \) combines a linear component with a rapidly oscillating sine function.
Key characteristics to remember include:

• Oscillation: The sine function oscillates between -1 and 1. When multiplied by \( x \), these oscillations result in an amplitude that approaches zero as \( x \to 0 \).
• Boundaries: Functions like sine are often used with bounding techniques such as the Squeeze Theorem, because while they may oscillate wildly, they have predictable upper and lower bounds.
• Applications: Beyond this problem, trigonometric functions are essential in modeling real-world phenomena such as wave patterns and circular motion.

Understanding these functions helps in interpreting behaviors and solving limits that involve rapid oscillations or wave-like properties.