To find the limit $\lim_{{x \to -2^-}} f(x)$, we need to analyze the behavior of the function as $x$ approaches $-2$ from the left side, denoted by the superscript $^-$. Based on your description of the graph and the equations of the curves, we can follow these steps:

Step 1: Identify the equation of the curve to the left of $x = -2$.

The graph in Region 1, which is the left region, corresponds to the function $f(x) = -2x + 2 - 1$. This equation defines the behavior of the function as $x$ approaches $-2$ from the left side.

Step 2: Evaluate the limit as $x$ approaches $-2$ from the left.

We will substitute $x = -2$ into the equation $f(x) = -2x + 2 - 1$:

$f(x) = -2(-2) + 2 - 1 = 4 + 2 - 1 = 5$

Conclusion:

As $x$ approaches $-2$ from the left ($x \to -2^-$), the value of $f(x)$ approaches $5$.

Thus, $\lim_{{x \to -2^-}} f(x) = 5$.

Would you like details on how to find other types of limits or clarify this one?

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Here are 5 related questions for further practice:

1. What is $\lim_{{x \to -2^+}} f(x)$ from the graph?
2. What is $\lim_{{x \to 2}} f(x)$ if there is a vertical asymptote?
3. How do we find $\lim_{{x \to 0}} f(x)$ if the function is continuous at $x = 0$?
4. How does the behavior of a graph near a vertical asymptote affect limits?
5. What is the difference between one-sided and two-sided limits?

Tip: Always pay attention to whether you are approaching a point from the left ($^-$) or from the right ($^+$) when finding one-sided limits.