1. Let's start by stating the problem: understanding the concept of limits in calculus. 2. A limit describes the value that a function $f(x)$ approaches as the input $x$ approaches some value $a$. It is written as $$\lim_{x \to a} f(x) = L$$ meaning as $x$ gets closer to $a$, $f(x)$ gets closer to $L$. 3. Important rules for limits include: - If $f(x)$ is continuous at $a$, then $$\lim_{x \to a} f(x) = f(a)$$. - Limits can be evaluated from the left ($x \to a^-$) or right ($x \to a^+$). - If the left and right limits are equal, the limit exists. 4. To find a limit, substitute $x = a$ into $f(x)$ if possible. If substitution leads to an indeterminate form like $\frac{0}{0}$, use algebraic simplification, factoring, or special limit rules. 5. Example: Find $$\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$$. - Substitute $x=2$: $$\frac{2^2 - 4}{2 - 2} = \frac{0}{0}$$ indeterminate. - Factor numerator: $$\frac{(x-2)(x+2)}{x-2}$$. - Cancel $(x-2)$: $$x+2$$. - Now substitute $x=2$: $$2 + 2 = 4$$. 6. So, $$\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = 4$$. This process helps us understand how functions behave near points even if they are not defined exactly at those points.