To solve the limit $\lim_{{x \to -2}} f(x)$ using the graph of $f(x)$, we need to examine the behavior of the function as $x$ approaches $-2$ from both the left and right.

Step 1: Approaching from the left ($x \to -2^-$)

As $x$ approaches $-2$ from the left side (values slightly less than $-2$), we follow the graph from the left. The curve appears to approach the point $(-2, -1)$. Thus, from the left-hand side, $f(x) \to -1$.

Step 2: Approaching from the right ($x \to -2^+$)

As $x$ approaches $-2$ from the right side (values slightly greater than $-2$), the graph also approaches the point $(-2, -1)$ from the right. Thus, from the right-hand side, $f(x) \to -1$.

Step 3: Verify if the limit exists

Since the left-hand limit and the right-hand limit both approach the same value $-1$, the overall limit exists.

Final Answer:

$\lim_{{x \to -2}} f(x) = -1$

Would you like further details or clarification?

Here are 5 related questions to expand your understanding:

1. How do you compute limits from a graph for piecewise functions?
2. What happens if the left-hand and right-hand limits differ?
3. How do you evaluate limits at points of discontinuity?
4. What are one-sided limits, and when are they useful?
5. How can the concept of limits help understand the derivative?

Tip: Always check both left-hand and right-hand limits when evaluating the existence of a limit at a particular point.