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A left-hand limit examines the behavior of a function as it approaches a certain value from the left side. For piecewise functions, we use the segment defined for values less than the point of interest when determining a left-hand limit. For instance, if we look at the exercise with function \(h(x)\), to calculate \(\lim_{x \rightarrow 2^{-}} h(x)\), we use the expression given for \(x < 2\), which is \(x^2 - 2x + 1\). By substituting \(x = 2\) into this quadratic equation, we derive:

• Substitution: \((2)^2 - 2 \times 2 + 1\)
• Calculation: \(4 - 4 + 1 = 1\)

Thus, the left-hand limit as \(x\) approaches 2 is 1. This process confirms continuity from the left even when the function switches formulas at \(x = 2\).

The right-hand limit is concerned with how a function behaves as it nears a certain value from the right side. For piecewise functions, we select the part defined for the specified range inclusive of or exceeding the point.In our example, to find \(\lim_{x \rightarrow 2^{+}} h(x)\), we apply the segment for \(x \geq 2\) which is \(3 - x\). We then substitute \(x = 2\) into this linear equation:

• Substitution: \(3 - 2\)
• Calculation: \(3 - 2 = 1\)

After calculating, we confirm \(\lim_{x \rightarrow 2^{+}} h(x) = 1\). By matching this result with the left-hand counterpart, we determine the function's behavior from both sides of \(x = 2\).

Continuity of a function at a certain point means its left-hand and right-hand limits are equal and the function value itself is defined and matches these limits. In a piecewise function, demonstrating continuity involves ensuring the behavior from both sides of a point converges.For the piecewise function \(h(x)\), the limits on both sides for \(x = 2\) are calculated as:

• Left-hand limit: \(1\)
• Right-hand limit: \(1\)

With both limits equating to 1, we see that \(h(2)\) also equals 1 when evaluated from either function component. Thus, continuity at \(x = 2\) is validated by the convergence of both directional limits with the function's actual value at this point. This ensures a seamless transition across the piecewise boundary at the evaluated point, indicating the function does not "break" at \(x = 2\).