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Piecewise functions are a type of function that are defined by different expressions or formulas depending on the input value. In simpler terms, it’s like a function that behaves differently in different parts of its domain. Consider it as a set of small functions clumped together, each piece active over a specific interval or range:

• These functions often have specific rules for different sections of the domain.
• The boundary points are especially important to check for continuity.

In our exercise, the function is defined as \( f(x) = \frac{\sin x}{|x|} \) when \( x eq 0 \), and equals 1 when \( x = 0 \).
This function changes its rule at \( x = 0 \), which creates a key point where continuity needs to be carefully evaluated. Understanding the behavior at different sections of the domain is crucial for analyzing functions like this one.

The limit of a function describes the behavior of the function as the input approaches a certain value. The limit is essentially what the function's output gets closer to as the input value approaches a specified point.

• It tells us how the function behaves as we near a particular point.
• A key concept for determining whether a function is continuous at a point.

In our problem, to check the continuity of \( f(x) \) at \( x = 0 \), we need to evaluate \( \lim_{{x \to 0}} f(x) \). This limit will show whether the function smoothly transitions through \( x = 0 \) or if there is a jump or undefined behavior.
Limits are fundamental in calculus as they allow for analysis of points that cannot be evaluated simply by substitution, like when dealing with piecewise functions.

Two-sided limits consider the behavior of a function as the input approaches a specific point from both the left and the right. For a limit at a point to exist, the function needs to approach the same value from either side of that point.

• The left-hand limit considers values approaching from the negative side.
• The right-hand limit considers values coming in from the positive side.

In our exercise, determining the two-sided limit of \( f(x) \) as \( x \) approaches 0 is crucial. From the positive side, the limit is \( 1 \), while from the negative side, it is \( -1 \).
Because these two results are not equal, the overall two-sided limit does not exist when \( x=0 \). This nonexistence of a consistent value signifies a break in the function's continuity at that point.