When Does a Two-Sided Limit Not Exist But One-Sided Limits Do?

📚 Understanding Limits: A Comprehensive Guide

In calculus, a limit describes the value that a function approaches as the input approaches some value. For a two-sided limit to exist at a point, the function must approach the same value from both the left and the right sides of that point. However, there are scenarios where the function approaches different values from each side, or doesn't approach any specific value at all, yet the one-sided limits exist. Let's delve into these scenarios.

📜 Historical Context

The concept of limits was formalized in the 19th century, primarily by mathematicians like Cauchy, Weierstrass, and Bolzano. They sought to provide a rigorous foundation for calculus, which had previously relied on more intuitive notions of infinitesimals. Understanding when limits exist—and when they don't—is crucial for this rigor.

📌 Key Principles

• 🔍 Definition of One-Sided Limits: The limit from the left (denoted as $ \lim_{x \to a^-} f(x) $) exists if $f(x)$ approaches a specific value as $x$ approaches $a$ from values less than $a$. Similarly, the limit from the right (denoted as $ \lim_{x \to a^+} f(x) $) exists if $f(x)$ approaches a specific value as $x$ approaches $a$ from values greater than $a$.
• 💡 Two-Sided Limit Existence: For the two-sided limit ($ \lim_{x \to a} f(x) $) to exist, both one-sided limits must exist and be equal, i.e., $ \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)$.
• 📝 Discontinuity Types: The non-existence of a two-sided limit while one-sided limits exist often occurs at points of discontinuity. Common types include jump discontinuities and essential discontinuities.

🌍 Real-World Examples

Let's explore some examples where a two-sided limit does not exist, but one-sided limits do:

1. Jump Discontinuity:
   Consider the Heaviside step function, defined as: $$ H(x) = \begin{cases} 0, & x < 0 \\ 1, & x \\geq 0 \end{cases} $$ Here, $ \lim_{x \to 0^-} H(x) = 0 $ and $ \lim_{x \to 0^+} H(x) = 1 $. Since the one-sided limits are not equal, the two-sided limit $ \lim_{x \to 0} H(x) $ does not exist.
2. Piecewise Functions:
   Consider a function defined as: $$ f(x) = \begin{cases} x^2, & x < 1 \\ 2x + 1, & x \\geq 1 \end{cases} $$ At $x = 1$, $ \lim_{x \to 1^-} f(x) = 1^2 = 1 $ and $ \lim_{x \to 1^+} f(x) = 2(1) + 1 = 3 $. The two-sided limit $ \lim_{x \to 1} f(x) $ does not exist because the one-sided limits differ.
3. Sign Function:
   The sign function is defined as: $$ \text{sgn}(x) = \begin{cases} -1, & x < 0 \\ 0, & x = 0 \\ 1, & x > 0 \end{cases} $$ At $x = 0$, $ \lim_{x \to 0^-} \text{sgn}(x) = -1 $ and $ \lim_{x \to 0^+} \text{sgn}(x) = 1 $. The two-sided limit $ \lim_{x \to 0} \text{sgn}(x) $ does not exist.

📊 Table Summary

🔑 Conclusion

The existence of one-sided limits without the existence of a two-sided limit is a critical concept in calculus, particularly when dealing with discontinuities. Understanding these scenarios provides a deeper insight into the behavior of functions and their limits. These examples and principles are fundamental for more advanced calculus topics.