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Piecewise functions are a special type of function that is defined by different expressions based on the value of the input variable, usually denoted by \( x \).
They often require evaluating expressions over different intervals.Understanding piecewise functions is crucial because:

• They allow complex functions to be expressed in simple parts.
• Each part deals with specific intervals of inputs.
• They can model real-world scenarios with varying conditions.

In the given exercise, we have a piecewise function \( f(x) \), which behaves differently based on whether \( x \) is less than, or greater than or equal to zero:

• For \( x < 0 \), \( f(x) = 2x \).
• For \( x \geq 0 \), \( f(x) = \frac{x}{2} \).

Each expression reflects a different rule or pattern of behavior depending on \( x \). Working with piecewise functions requires careful attention to which part of the function applies to the value of \( x \) being considered.

The left-hand limit focuses on what happens to a function as the input \( x \) approaches a specific point from the left side. This means for values slightly less than the point in question. For our piecewise function, we want to examine:\[ \lim_{x \to 0^-} f(x) \]Here, \( x \to 0^- \) indicates \( x \) approaching zero from values less than zero. For the piecewise function given, we consider the piece where \( x < 0 \).
Thus, the relevant expression is \( f(x) = 2x \).To evaluate the left-hand limit:

• We substitute \( 2x \) into the limit expression: \( \lim_{x \to 0^-} 2x \).
• Calculating this gives us: \( 2 \times 0 = 0 \).

This calculation shows that as \( x \) approaches zero from the left, the value of the function approaches 0. It is vital in limit analysis, especially for verifying limits at specific points.

The right-hand limit considers what happens to a function as \( x \) approaches a specific point from the right side, or from slightly larger values than the point itself. For our exercise, this means exploring:\[ \lim_{x \to 0^+} f(x) \]The notation \( x \to 0^+ \) signifies \( x \) approaching zero from the right. We need to look at the interval where \( x \geq 0 \) for our piecewise function, using the part \( f(x) = \frac{x}{2} \).Steps to find the right-hand limit include:

• Substitute \( \frac{x}{2} \) for \( f(x) \) in the limit: \( \lim_{x \to 0^+} \frac{x}{2} \).
• Evaluate to get \( \frac{0}{2} = 0 \).

This confirms that as \( x \) approaches zero from the right, \( f(x) \) also approaches 0. Matching left-hand and right-hand limits is essential to ensuring the overall limit exists.

The limit process involves determining the value that a function approaches as the input approaches a particular point. It's a foundational concept in calculus and provides essential understanding of function behavior.In this exercise, the limit process looks at:

• The limit as \( x \) approaches zero from the left, yielding \( \lim_{x \to 0^-} f(x) = 0 \).
• The limit as \( x \) approaches zero from the right, giving \( \lim_{x \to 0^+} f(x) = 0 \).
• Ensuring both these one-sided limits are equal confirms \( \lim_{x \to 0} f(x) = 0 \).

The limit process not only checks these individual limits but also ensures continuity at the point if both match. The overall approach guarantees the piecewise function smoothly transitions between its parts at the given point, confirming that a limit indeed exists. This deepens understanding of function behavior beyond just calculating limit values.