How to evaluate limits of piecewise functions algebraically

📚 Evaluating Limits of Piecewise Functions Algebraically

Piecewise functions are functions defined by multiple sub-functions, each applying to a certain interval of the main function's domain. Evaluating limits of piecewise functions requires careful consideration of which sub-function applies as you approach a specific x-value. Let's break it down:

📜 History and Background

The concept of piecewise functions has been around implicitly since the early days of calculus, arising in problems where different rules governed different parts of a physical system or geometric shape. The formal notation and treatment of these functions developed gradually as mathematical notation became standardized.

🔑 Key Principles

• 🔍Definition: A piecewise function is a function defined by multiple sub-functions on different intervals of its domain. For example: $f(x) = \begin{cases} x^2, & x < 0 \\ x + 1, & x \geq 0 \end{cases}$
• ➡️One-Sided Limits: To evaluate the limit as $x$ approaches a value $c$, we must consider the left-hand limit ($x \to c^-$) and the right-hand limit ($x \to c^+$).
• ✔️Matching Limits: The limit $\lim_{x \to c} f(x)$ exists if and only if both the left-hand limit and the right-hand limit exist and are equal: $\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x)$.
• ⛔Discontinuities: Piecewise functions are often used to model situations with discontinuities. At the points where the function changes definition, the limit may not exist.

💡 Real-World Examples

Piecewise functions are used in various applications such as:

• 📊Tax Brackets: The amount of tax you pay often depends on your income bracket, which is a piecewise function.
• 🌡️Thermostat Control: A thermostat might use different rules to heat or cool a room based on the current temperature.
• 🚦Traffic Flow: Traffic patterns can be modeled with different functions depending on the time of day.

📝 Example 1:

Let's consider the function:

We want to find $\lim_{x \to 1} f(x)$.

• ⬅️Left-hand limit: $\lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} (x + 2) = 1 + 2 = 3$.
• ➡️Right-hand limit: $\lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} (3x) = 3(1) = 3$.
• ✅Conclusion: Since the left-hand limit and right-hand limit are equal, $\lim_{x \to 1} f(x) = 3$.

🧪 Example 2:

Now, consider the function:

We want to find $\lim_{x \to 2} g(x)$.

• ⬅️Left-hand limit: $\lim_{x \to 2^-} g(x) = \lim_{x \to 2^-} (x^2) = (2)^2 = 4$.
• ➡️Right-hand limit: $\lim_{x \to 2^+} g(x) = \lim_{x \to 2^+} (2x + 1) = 2(2) + 1 = 5$.
• ❌Conclusion: Since the left-hand limit and right-hand limit are not equal, $\lim_{x \to 2} g(x)$ does not exist.

📝 Example 3: A More Complex Case

Let's look at a trigonometric piecewise function:

Find the value of $a$ such that $\lim_{x \to 0} h(x)$ exists.

• ⬅️Left-hand limit: $\lim_{x \to 0^-} h(x) = \lim_{x \to 0^-} \sin(x) = \sin(0) = 0$.
• ➡️Right-hand limit: $\lim_{x \to 0^+} h(x) = \lim_{x \to 0^+} (x + a) = 0 + a = a$.
• 🧮Solving for a: For the limit to exist, $0 = a$. Therefore, $a = 0$.

🎯 Conclusion

Evaluating limits of piecewise functions involves checking the one-sided limits at the points where the function's definition changes. If the left-hand and right-hand limits are equal, the limit exists at that point. If they are not equal, the limit does not exist. Piecewise functions provide a flexible way to model a variety of real-world phenomena where different rules apply in different situations.