How to interpret one-sided limits from a given graph

📚 Understanding One-Sided Limits from Graphs

One-sided limits help us understand what happens to a function as it approaches a particular $x$-value from either the left or the right side. Unlike a regular limit, which requires the function to approach the same value from both sides, one-sided limits focus on behavior from a single direction. Let's break it down!

📜 History and Background

The concept of limits, including one-sided limits, is fundamental to calculus. It was developed rigorously in the 19th century to formalize ideas about continuity, derivatives, and integrals. Mathematicians like Cauchy and Weierstrass played key roles in defining limits precisely.

🔑 Key Principles

• ➡️ Right-Hand Limit: The limit as $x$ approaches $a$ from the right (denoted as $x \to a^+$) examines the function's behavior for $x$ values slightly greater than $a$.
• ⬅️ Left-Hand Limit: The limit as $x$ approaches $a$ from the left (denoted as $x \to a^-$) looks at the function's behavior for $x$ values slightly less than $a$.
• 🤝 Existence of a Limit: For a regular limit to exist at $x = a$, both the left-hand limit and the right-hand limit must exist and be equal. If they are not equal, the limit does not exist at that point.
• 🛑 Discontinuities: One-sided limits are particularly useful when dealing with discontinuities, such as jump discontinuities, where the function abruptly changes value.

📈 Interpreting from a Graph

To interpret one-sided limits from a graph:

• 👀 Identify the Point of Interest: Find the $x$-value, $a$, at which you want to evaluate the one-sided limits.
• 👉 Right-Hand Limit: Trace the graph from the right side towards $x = a$. The $y$-value that the graph approaches is the right-hand limit.
• 👈 Left-Hand Limit: Trace the graph from the left side towards $x = a$. The $y$-value that the graph approaches is the left-hand limit.
• 💥 Compare: If the left-hand and right-hand limits are different, the overall limit at $x = a$ does not exist.

🌍 Real-World Examples

Consider a function representing the cost of shipping, where the cost changes at specific weight intervals.

Let's say the cost function is defined as follows:

Here, $C(w)$ is the cost to ship a package of weight $w$ (in pounds). What happens at $w = 1$?

• 📦 Limit from the Left: $\lim_{w \to 1^-} C(w) = 5$. As the weight approaches 1 pound from below, the cost approaches $5.
• 🚚 Limit from the Right: $\lim_{w \to 1^+} C(w) = 8$. As the weight approaches 1 pound from above, the cost approaches $8.
• 🚧 Conclusion: Since the left and right limits are not equal, the overall limit at $w = 1$ does not exist.

📊 Example with a Piecewise Function

Consider the piecewise function:

• 🔍 Left-Hand Limit at $x = 2$: $\lim_{x \to 2^-} f(x) = (2)^2 = 4$.
• 💡 Right-Hand Limit at $x = 2$: $\lim_{x \to 2^+} f(x) = 3(2) - 1 = 5$.
• 📝 Overall Limit: Since $4 \neq 5$, the limit $\lim_{x \to 2} f(x)$ does not exist.

🧩 Conclusion

Understanding one-sided limits is essential for analyzing function behavior, especially at points of discontinuity. By examining the limits from the left and right, you gain a complete picture of how a function behaves around a specific point. This knowledge is crucial for calculus and real-world applications involving piecewise functions and discontinuous processes.