1. Let's start by understanding the problem: you want to learn about functions and limits. 2. A function is a relation where each input has exactly one output. For example, $f(x) = 2x + 3$ means for every $x$, the output is $2x + 3$. 3. The limit of a function at a point tells us what value the function approaches as the input gets closer to that point. 4. The formal definition of a limit is: $$\lim_{x \to a} f(x) = L$$ means as $x$ approaches $a$, $f(x)$ approaches $L$. 5. Important rules for limits include: - Limits can be evaluated by direct substitution if the function is continuous at that point. - If direct substitution leads to an indeterminate form like $\frac{0}{0}$, we simplify the function. 6. Example: Find $$\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$$ 7. Direct substitution gives $$\frac{2^2 - 4}{2 - 2} = \frac{0}{0}$$ which is indeterminate. 8. Factor numerator: $$x^2 - 4 = (x - 2)(x + 2)$$ 9. So the expression becomes $$\frac{(x - 2)(x + 2)}{x - 2}$$ 10. Cancel common factor $x - 2$: $$\frac{\cancel{(x - 2)}(x + 2)}{\cancel{(x - 2)}} = x + 2$$ 11. Now substitute $x = 2$: $$2 + 2 = 4$$ 12. Therefore, $$\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = 4$$. This shows how limits help us find values even when direct substitution initially fails.