Let's analyze the graph to answer each of the given limits and function values.

Given Information

We need to estimate values for:

1. $\lim_{{x \to -2^-}} f(x)$
2. $\lim_{{x \to -2^+}} f(x)$
3. $\lim_{{x \to -2}} f(x)$
4. $f(-2)$
5. $\lim_{{x \to \infty}} f(x)$

Based on the graph, let's proceed with each part.

Solutions

(a) $\lim_{{x \to -2^-}} f(x)$:

• As $x$ approaches $-2$ from the left side ($x \to -2^-$), the value of $f(x)$ approaches approximately $1$.

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

(b) $\lim_{{x \to -2^+}} f(x)$:

• As $x$ approaches $-2$ from the right side ($x \to -2^+$), $f(x)$ also appears to approach $1$.

Answer: $\lim_{{x \to -2^+}} f(x) = 1$

(c) $\lim_{{x \to -2}} f(x)$:

• Since both the left-hand limit and the right-hand limit as $x \to -2$ approach $1$, the two-sided limit also equals $1$.

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

(d) $f(-2)$:

• From the graph, it appears there is a filled dot at $(-2, 1)$, indicating $f(-2) = 1$.

Answer: $f(-2) = 1$

(e) $\lim_{{x \to \infty}} f(x)$:

• As $x$ approaches infinity ($x \to \infty$), $f(x)$ seems to be decreasing towards $0$.

Answer: $\lim_{{x \to \infty}} f(x) = 0$

Summary of Answers

1. $\lim_{{x \to -2^-}} f(x) = 1$
2. $\lim_{{x \to -2^+}} f(x) = 1$
3. $\lim_{{x \to -2}} f(x) = 1$
4. $f(-2) = 1$
5. $\lim_{{x \to \infty}} f(x) = 0$

Would you like a more detailed explanation of any part?

Here are some related questions to expand your understanding:

1. What is the significance of one-sided limits in determining two-sided limits?
2. How does the existence of a point at $x = -2$ affect the value of the limit?
3. What happens to the limit if the function approaches different values from the left and right?
4. How would you interpret a limit as $x \to -\infty$ for the function $f(x)$?
5. Why is it important to observe both the left-hand and right-hand limits for continuity?

Tip: When estimating limits from a graph, always check if the function approaches the same value from both sides for a two-sided limit!