Use the graph of the function f to state the value of each limit, if it exists. (If an answer does not exist, enter DNE.)
f(x) = x

2 + x−2
(a)
lim
x→0−
f(x)
(b)
lim
x→0+
f(x)
(c)
lim
x→0
f(x)

Let's analyze the function \( f(x) = \frac{2 + x}{x - 2} \) to determine the values of the limits at \( x = 0 \).

### Function Analysis:
\[ f(x) = \frac{2 + x}{x - 2} \]

#### (a) \( \lim_{x \to 0^-} f(x) \)
This is the left-hand limit as \( x \) approaches 0 from the left. Since \( x - 2 \) is negative for \( x < 2 \), the function is defined in this region. We can substitute values slightly less than 0 into the function:

\[ \lim_{x \to 0^-} f(x) = \frac{2 + 0^-}{0^- - 2} = \frac{2}{-2} = -1 \]

So:
\[ \lim_{x \to 0^-} f(x) = -1 \]

#### (b) \( \lim_{x \to 0^+} f(x) \)
This is the right-hand limit as \( x \) approaches 0 from the right. Similarly, for \( x > -2 \), \( x - 2 \) is negative, and we substitute values slightly greater than 0:

\[ \lim_{x \to 0^+} f(x) = \frac{2 + 0^+}{0^+ - 2} = \frac{2}{-2} = -1 \]

So:
\[ \lim_{x \to 0^+} f(x) = -1 \]

#### (c) \( \lim_{x \to 0} f(x) \)
The limit exists only if the left-hand and right-hand limits are equal:

\[ \lim_{x \to 0} f(x) = -1 \]

### Final Answers:
(a) \( \lim_{x \to 0^-} f(x) = -1 \)

(b) \( \lim_{x \to 0^+} f(x) = -1 \)

(c) \( \lim_{x \to 0} f(x) = -1 \)

Would you like further details or have any questions?

### Related Questions:
1. What is the limit \( \lim_{x \to 2^-} f(x) \)?
2. How do vertical asymptotes affect limits of functions?
3. Can a function have different left-hand and right-hand limits?
4. What is the significance of continuity in determining limits?
5. How do you compute limits for functions involving absolute values?

**Tip:** When evaluating limits, always consider the function's behavior from both the left and right sides of the point of interest.

Learn how to evaluate the limits of the function f(x) = (2 + x) / (x - 2) as x approaches 0 from the left, right, and overall. Understand the concepts of left-hand limit, right-hand limit, and continuity in functions. Suitable for high school students studying calculus.

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