Piecewise Graph Limit Calculation
The image presents a piecewise graph of the function y = f(x) and asks for the value of the expression:
\[ \lim_{x \to -4^+} f(x) - \lim_{x \to 0^-} f(x) + \lim_{x \to 2^+} f(x) \]
From the graph, we can derive the following limits:
1. \( \lim_{x \to -4^+} f(x) \) is the value that f(x) approaches as x approaches -4 from the right. By looking at the graph, we can see this value is n.
2. \( \lim_{x \to 0^-} f(x) \) is the value that f(x) approaches as x approaches 0 from the left. On the graph, this value is m.
3. \( \lim_{x \to 2^+} f(x) \) is the value that f(x) approaches as x approaches 2 from the right. The graph shows this value as k.
Now, we substitute these values into the given expression:
n (the limit as x approaches -4 from the right) - m (the limit as x approaches 0 from the left) + k (the limit as x approaches 2 from the right).
From the graph, we can infer the following based on the labelled points:
- n is at y = -1
- m is at y = 1
- k is at y = 0
Using these values:
\( \lim_{x \to -4^+} f(x) = -1 \)
\( \lim_{x \to 0^-} f(x) = 1 \)
\( \lim_{x \to 2^+} f(x) = 0 \)
Substitute these values into the expression:
\[ (-1) - (1) + (0) = -2 \]
Therefore, the value of the given expression is -2.
