Understanding Limit 1D23D8

1. Let's start by stating the problem: We want to understand what a limit is in mathematics. 2. A limit describes the value that a function $f(x)$ approaches as the input $x$ approaches some value $a$. 3. The formal notation is $$\lim_{x \to a} f(x) = L$$ which means as $x$ gets closer and closer to $a$, $f(x)$ gets closer and closer to $L$. 4. Important rules: - The limit may exist even if $f(a)$ is not defined. - The limit depends on the behavior of $f(x)$ near $a$, not necessarily at $a$. 5. Example: Consider $f(x) = \frac{x^2 - 1}{x - 1}$ and find $$\lim_{x \to 1} f(x)$$. 6. Simplify the function: $$f(x) = \frac{x^2 - 1}{x - 1} = \frac{(x - 1)(x + 1)}{x - 1}$$ 7. Cancel the common factor: $$= \frac{\cancel{(x - 1)}(x + 1)}{\cancel{(x - 1)}} = x + 1$$ 8. Now evaluate the limit by substituting $x = 1$: $$\lim_{x \to 1} f(x) = 1 + 1 = 2$$ 9. So, even though $f(1)$ is undefined (division by zero), the limit as $x$ approaches 1 is 2. 10. This shows how limits help us understand function behavior near points where the function might not be defined.