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A piecewise function is a type of function that is defined by different expressions depending on the input value. Think of it as a function with multiple "pieces" that come together to form a complete mathematical description. Each piece of the function applies to certain parts of the domain.
For the function given in the exercise, it is defined as follows:

• When \( x eq 0 \), it follows the rule \( x^4 \sin(1/x) \).
• When \( x = 0 \), it uses the rule \( f(x) = 0 \).

This setup allows the function to have different behaviors at different points, which is especially useful for managing discontinuities or complex functions that change behavior at specific points.
To understand piecewise functions better, remember that each part is continuous in its own domain, and special attention is required to check the points where the rules change, such as \( x=0 \) in this case.

The Squeeze Theorem is a powerful mathematical tool used primarily to find limits. It comes into play especially when direct evaluation is challenging. The idea is simple: if a function \( f(x) \) is sandwiched between two other functions that have the same limit at a point, then \( f(x) \) must also share that limit.
In the exercise, the function \( x^4 \sin(1/x) \) is tricky to handle directly as \( x \) approaches zero. However, we know that \( \sin(1/x) \) is always between -1 and 1. Therefore, the product \( x^4 \sin(1/x) \) is between \( -x^4 \) and \( x^4 \).
This is where the Squeeze Theorem shines:

• As \( x \to 0 \), both \( \lim_{x \to 0} -x^4 = 0 \) and \( \lim_{x \to 0} x^4 = 0 \).

Therefore, by squeezing \( x^4 \sin(1/x) \) between these two expressions, we conclude \( \lim_{x \to 0} x^4 \sin(1/x) = 0 \).
This conclusion is crucial to show that the function is continuous at \( x = 0 \), as required.

Limits are foundational in calculus, providing a way to analyze the behavior of functions as inputs approach a particular value. To check whether our piecewise function is continuous, we need to evaluate limits, especially at \( x = 0 \).
The limit of a function \( \lim_{x \to c} f(x) = L \) tells us that as \( x \) gets closer and closer to some number \( c \), \( f(x) \) approaches \( L \). For continuity, it's not enough for \( f(x) \) to approach \( L \), it must equal \( f(c) \).
In the given exercise, we determined:

• \( \lim_{x \to 0} x^4 \sin(1/x) = 0 \).
• We needed this limit to equal \( f(0) = 0 \).

Since both are 0, the function is continuous at \( x = 0 \). This satisfies a crucial part of verifying continuity across the entire domain of a piecewise function.

Sinusoidal functions are functions that involve sine or cosine. These functions are periodic and oscillate between values, creating continuous waves. In our exercise, the function involves \( \sin(1/x) \).
The sine function has some straightforward properties that make it predictable:

• It oscillates between -1 and 1 for all inputs, which aids in determining bounds for more complex functions.
• It's continuous everywhere by nature.

In the piecewise function of the exercise, \( \sin(1/x) \) is multiplied by \( x^4 \), which causes its oscillation behavior to affect the overall function.
The bounding property of sine was critical in applying the Squeeze Theorem since \( -1 \leq \sin(1/x) \leq 1 \) implied that \( -x^4 \leq x^4 \sin(1/x) \leq x^4 \). Understanding the role of sinusoidal functions helps in evaluating how oscillations impact function behavior near critical points like \( x = 0 \).