Key Differences Between One-Sided and Two-Sided Limits

1. What They Mean:

   • One-Sided Limit: This is when we look at what value a function gets close to as we come from one side:
     • Left-Hand Limit: This is written as $\lim_{x \to a^-} f(x)$. It shows the value as we approach from the left.
     • Right-Hand Limit: This one is written as $\lim_{x \to a^+} f(x)$. It shows the value as we come from the right.
   • Two-Sided Limit: This is when we check both sides. The limit exists if both one-sided limits give the same number:
     • We write it as $\lim_{x \to a} f(x) = L$ if $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L$$.

2. When They Exist:

   • A two-sided limit won’t exist if the one-sided limits are different. For example, if we find that $\lim_{x \to a^-} f(x) = 2$ and $\lim_{x \to a^+} f(x) = 3$, then $\lim_{x \to a} f(x)$ does not exist.

3. How We Write Them:

   • We use special notes to show one-sided limits. A superscript (like $^-$ for left or $^+$ for right) tells you which side we mean. Two-sided limits are written simply without showing direction.

Knowing these differences helps us understand limits better and makes it easier to check if functions are continuous.