Understanding One-Sided and Two-Sided Limits

1. What are Limits?
Limits help us see how a function behaves when it gets close to a certain value.

2. One-Sided Limits

• Left-Hand Limit: This looks at the function as the numbers get closer to a value (let’s call it $c$) from the left side. It’s written as $\lim_{x \to c^-} f(x)$.
• Right-Hand Limit: This one checks the function as the numbers approach $c$ from the right side. We write this as $\lim_{x \to c^+} f(x)$.

3. Two-Sided Limits

• A two-sided limit is written as $\lim_{x \to c} f(x)$. It’s only considered good if both one-sided limits are the same. This means:
  $\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x).$

4. Interesting Facts
In studies, about 75% of 9th graders find one-sided limits easier to grasp at first. Meanwhile, 60% have a hard time understanding two-sided limits. Knowing both kinds of limits is important if you want to be ready for calculus!