Piecewise Functions

Concept Explanation

Piecewise functions model systems whose underlying physics or operating rules change depending on the input condition. Examples include tax brackets, step voltages, traffic lights, and structural loads along non-uniform beams.

Evaluating Limits of Piecewise Functions

To evaluate $\lim_{x \to a} f(x)$ for a piecewise function:

1. Interior Points: If $a$ lies strictly inside one sub-interval (e.g. $a < c$), evaluate the limit using the corresponding sub-function rule directly.
2. Boundary Points: If $a$ is a transition threshold between two pieces (e.g. $x = c$), you must evaluate both directional limits:
   • Left-hand limit: $\lim_{x \to c^-} f(x)$ using the formula for $x < c$.
   • Right-hand limit: $\lim_{x \to c^+} f(x)$ using the formula for $x > c$.
3. Existence Check: Apply the Existence Theorem:
   • If $\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = L$, then $\lim_{x \to c} f(x) = L$.
   • If they disagree, the limit is DNE due to a Jump Discontinuity.

---

Example Problems

Example Problem 1: Three-Piece Piecewise Function at Boundaries

Consider the function:

Evaluate: (a) $\lim_{x \to -5} f(x)$ and (b) $\lim_{x \to 4} f(x)$.

Solution

Part (a) — Boundary at $x = -5$

• Left limit ($x < -5$): $\lim_{x \to -5^-} (0) = 0$
• Right limit ($x > -5$): $\lim_{x \to -5^+} \sqrt{25 - (-5)^2} = \sqrt{0} = 0$
• Since both one-sided limits equal $0$:
  
  $$\lim_{x \to -5} f(x) = 0$$

Part (b) — Boundary at $x = 4$

• Left limit ($x < 4$): $\lim_{x \to 4^-} \sqrt{25 - (4)^2} = \sqrt{9} = 3$
• Right limit ($x > 4$): $\lim_{x \to 4^+} (5x) = 5(4) = 20$
• Since $3 \neq 20$, the directional limits disagree:
  
  $$\lim_{x \to 4} f(x) = \text{DNE} \quad (\text{Jump Discontinuity})$$

Example Problem 2: Function Value vs Limit Value in Piecewise Definitions

Evaluate $\lim_{x \to 1} g(x)$ and $g(1)$ for:

Solution

Step 1: Left limit ($x \to 1^-$)

$$\lim_{x \to 1^-} (2x + 1) = 2(1) + 1 = 3$$

Step 2: Right limit ($x \to 1^+$)

$$\lim_{x \to 1^+} (x^2 + 2) = 1^2 + 2 = 3$$

Step 3: Compare with function value

• Limit: $\lim_{x \to 1} g(x) = 3$ (because both sides agree).
• Function value: $g(1) = 6$.

Conclusion: The value of $g(1) = 6$ does not alter the limit value $\lim_{x \to 1} g(x) = 3$. The function exhibits a removable discontinuity (hole) at $(1, 3)$.

---

Connections & References

• Parent Concepts: Limits, Calculus Foundations
• Related Concepts: Existence Theorem for Two-Sided Limits, Strategies for One-Sided Limits, Discontinuities, Jump Discontinuity
• Course Links: Calc-1 Session 03