1. Let's start with the basic idea: A limit in calculus describes the value that a function approaches as the input approaches some value. 2. For example, consider the function $f(x) = 2x$. What happens to $f(x)$ as $x$ gets closer to 3? 3. We calculate $\lim_{x \to 3} 2x$. As $x$ approaches 3, $2x$ approaches 6. 4. This means the limit of $2x$ as $x$ approaches 3 is 6. 5. More formally, we write: $$\lim_{x \to a} f(x) = L$$ which means as $x$ gets arbitrarily close to $a$, $f(x)$ gets arbitrarily close to $L$. 6. Limits help us understand behavior of functions near points where they might not be explicitly defined, or where evaluating directly is difficult. 7. For example, consider $f(x) = \frac{x^2 - 9}{x - 3}$. Direct substitution at $x=3$ gives division by zero. 8. But factoring numerator gives $f(x) = \frac{(x-3)(x+3)}{x-3}$. For $x \neq 3$, $f(x) = x + 3$. 9. So, $\lim_{x \to 3} \frac{x^2 - 9}{x - 3} = \lim_{x \to 3} (x + 3) = 6$. 10. Thus, even though $f(3)$ is undefined, the limit as $x$ approaches 3 exists and equals 6. 11. Limits can also describe infinite behavior, continuity, and are fundamental for derivatives and integrals in calculus.