91Ó°ÊÓ

In the context of functions, a limit describes the value that a function approaches as the input variable approaches a certain point. Understanding limits is crucial when working with piecewise functions like \(h(x)\). Here's how we can approach the limits for \(h(x)\):
For \(\lim_{x \to 0^{+}} h(x)\), when \(x\) approaches zero from the positive side, we use \(h(x) = x^2\) since \(0 < x \leq 2\). As \(x\) becomes very small but positive, the function approaches zero. Hence, the limit is 0.
When evaluating \(\lim_{x \to 0} h(x)\), we consider both directions: left and right. From the left \((x \to 0^{-})\), \(h(x)\) equals \(x\), which approaches 0. From the right, already we've noted that \(h(x) = 0\). Since both approaches agree, \(\lim_{x \to 0} h(x) = 0\).
For \(\lim_{x \to 2^{-}} h(x)\), approaching 2 from the left means using \(h(x) = x^2\), and thus the limit dequals 4 since \(2^2 = 4\).
For \(\lim_{x \to 2^{+}} h(x)\), approaching 2 from the right involves \(h(x) = 8 - x\), resulting in the limit of 6, as \(8 - 2 = 6\).
Since \(\lim_{x \to 2^{-}}\) and \(\lim_{x \to 2^{+}}\) don't match, \(\lim_{x \to 2} h(x)\) doesn't exist.

Graphing piecewise functions can initially seem complex due to their segmented nature. However, by plotting each segment within its applicable range, you can visualize the function's behavior. For the function \(h(x)\) given here:

• For \(x < 0\), the function is linear: \(y = x\). This graph is a straight line passing through the origin with a 45-degree angle.
• For \(0 < x \leq 2\), the piece \(y = x^2\) is a segment of a parabola. It starts right after \(x = 0\) and ends at \(x = 2\), forming a smooth curve.
• Finally, for \(x > 2\), \(y = 8 - x\) is again linear, but with a negative slope. The line keeps decreasing as \(x\) increases.

Understanding the graph requires noting the transitions: smoothly meeting at \(x = 0\), and a jump at \(x = 2\), because the left limit and right limit differ.

Continuity refers to the function being unbroken or having no jumps. For piecewise functions, checking continuity at transition points is essential. For \(h(x)\), consider these points:

• At \(x = 0\), the function smoothly transitions from \(y = x\) to \(y = x^2\). The limit from both sides is zero, ensuring continuity.
• At \(x = 2\), there's a different story. Here, the left-hand limit \(\lim_{x \to 2^{-}}\) is 4, while the right-hand limit \(\lim_{x \to 2^{+}}\) is 6. Because these values differ, \(h(x)\) is discontinuous at \(x = 2\).

A function is continuous when it has no breaks, holes, or jumps, and \(h(x)\) is continuous everywhere except at \(x = 2\).

Parabolic functions are equations of the form \(y = ax^2 + bx + c\). In our piecewise example, \(h(x)\) includes a segment \(y = x^2\) for \(0 < x \leq 2\), which is a classic parabola.
Parabolas have a characteristic U-shape. For \(h(x)\), this part graphically denotes a steady increase as \(x\) increases from just above 0 to 2. This segment contributes to understanding how \(h(x)\) behaves at different domains.
When graphing a parabola:

• The vertex is a crucial point—here at \((0, 0)\).
• The axis of symmetry for this piece is the y-axis, given \(b = 0\) in \(ax^2 + bx + c\).
• Only part of the parabola is visible (from \(0\) to \(2\)), reflecting its piecewise nature.

Knowing parabolic functions allows us to predict how functions might curve and find their equations from graphs. It is a key skill when analyzing any segment of parabolic nature.