1. **State the problem:** We are given a piecewise function $$f(x) = \begin{cases} 3, & x \leq -1 \\ (x-1)^2, & x > -1 \end{cases}$$ We need to evaluate the left-hand limit as $x$ approaches $-1$, i.e., $\lim_{x \to -1^-} f(x)$. 2. **Recall the definition of left-hand limit:** The left-hand limit $\lim_{x \to a^-} f(x)$ is the value that $f(x)$ approaches as $x$ approaches $a$ from values less than $a$. 3. **Apply the definition to our function:** For $x \leq -1$, $f(x) = 3$. So as $x$ approaches $-1$ from the left, $f(x)$ is constantly 3. 4. **Evaluate the limit:** Since $f(x) = 3$ for all $x \leq -1$, $$\lim_{x \to -1^-} f(x) = 3.$$