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Piecewise functions are special types of functions defined by different expressions depending on the region of the input variable, typically denoted as "x." They allow us to construct functions that can exhibit different behaviors over various intervals. In the case of the piecewise function given by:

• For values of x less than 0, the function is defined as \( f(x) = 2x^2 \).
• For values of x greater than or equal to 0, the function is defined as \( f(x) = 4x \).

This means the function changes its rule at the point where x equals 0. Understanding piecewise functions is crucial because they appear in various real-world scenarios where a relationship changes at a certain value of the input, like tax brackets or speed-dependent travel times. Analyzing a piecewise function requires examining each part of the function separately within its specified interval to fully understand its behavior.

Limits are foundational in understanding calculus and analysis, as they explain the behavior of functions as inputs approach a certain value. A limit tells us what value a function approaches as the input gets infinitely close to some point. In our exercise, we looked at limits approaching 0 from both negative and positive directions.

For \( \lim_{x \rightarrow 0^-} f(x) \), we check the behavior of \( f(x) \) as x approaches 0 from the left. Since the expression for x < 0 is \( 2x^2 \), the limit is calculated as \( \lim_{x \rightarrow 0^-} 2x^2 = 0 \).

Similarly, for \( \lim_{x \rightarrow 0^+} f(x) \), we observe from the right. With the defined expression for x ≥ 0 being \( 4x \), the limit becomes \( \lim_{x \rightarrow 0^+} 4x = 0 \).

Understanding limits helps us to comprehend how functions behave near points of interest and to establish continuity and other crucial properties of functions.

Evaluating limits is a key step in determining if a piecewise function is continuous at a certain point. Continuity at a particular x-value means that the function not only reaches a limit from both sides but also that these limit values and the actual function evaluation at the point itself are equal.

For the function in the exercise:

• The left-hand limit \( \lim_{x \rightarrow 0^-} f(x) = 0 \)
• The function's value at 0 is \( f(0) = 0 \)
• The right-hand limit \( \lim_{x \rightarrow 0^+} f(x) = 0 \)

All these conditions - left-hand limit, right-hand limit, and the function value - meet at 0, which confirms the function is continuous at \( x = 0 \). Detecting these equalities emphasizes the balance of the function behavior as it transitions through a critical point, like \( x = 0 \) in this case.