1. The problem asks to define the term limit $L$ of a continuous function $f(x)$ about a point $x=p$. 2. The limit of a function $f(x)$ as $x$ approaches $p$ is the value that $f(x)$ gets closer to as $x$ gets closer to $p$. 3. Formally, we say \textit{the limit of $f(x)$ as $x$ approaches $p$ is $L$} if for every number $\epsilon > 0$, there exists a number $\delta > 0$ such that whenever $0 < |x - p| < \delta$, it follows that $|f(x) - L| < \epsilon$. 4. In simpler terms, this means that by choosing $x$ values sufficiently close to $p$ (but not equal to $p$), the values of $f(x)$ can be made as close as we want to $L$. 5. For a function to be continuous at $x=p$, the limit of $f(x)$ as $x$ approaches $p$ must exist and be equal to $f(p)$, i.e., $\lim_{x \to p} f(x) = f(p) = L$. 6. Therefore, the limit $L$ represents the value that the function approaches near the point $p$, ensuring no sudden jumps or breaks at $p$ for continuous functions.