#AP Calculus AB/BC: Estimating Limits from Tables 📈

Welcome to your ultimate guide for mastering limits using tables! This guide is designed to be your go-to resource, especially the night before the exam. Let's make sure you're confident and ready to ace it!

#Concept of a Limit Recap

As we've discussed, a limit describes the behavior of a function as its input approaches a certain value. The notation looks like this:

$$\lim_{x \to a} f(x) = L$$

Here, 'x' approaches 'a', and 'L' is the limit. We're diving into how to find 'L' using tables. 🔢

#One-Sided Limits

Before we jump into tables, it's crucial to understand one-sided limits. These look at the function's behavior as 'x' approaches 'a' from either the left (values less than 'a') or the right (values greater than 'a').

• Approaching from the right (values greater than 'a'): $$\lim_{x \to a^+} f(x) = L$$
• Approaching from the left (values less than 'a'): $$\lim_{x \to a^-} f(x) = L$$

Think of it like approaching a destination from different directions. ⬅️➡️

#Using a Table to Estimate Limits

Sometimes, direct substitution doesn't work (we get an indeterminate form like $\frac{0}{0}$). In these cases, we use a table to see what happens as 'x' gets close to 'a'. ❌

For example:

$$\lim_{x \to 0} \frac{x}{\sqrt{x+1}-1}$$

Direct substitution gives us $\frac{0}{0}$, which is und...