Understanding Limits of a Function from a Graph
The image shows a graph of a function y = f(x) and asks for the values of various limits as x approaches different numbers:
a) \( \lim_{x \to -2^-} f(x) \)
b) \( \lim_{x \to -2^+} f(x) \)
c) \( \lim_{x \to -2} f(x) \)
d) \( \lim_{x \to 3} f(x) \)
e) \( \lim_{x \to +\infty} f(x) \)
Based on the graph depicted in the image:
a) \( \lim_{x \to -2^-} f(x) \) is the limit of the function as x approaches -2 from the left. The graph shows that as x approaches -2 from the left, y approaches -1. So the limit is -1.
b) \( \lim_{x \to -2^+} f(x) \) is the limit of the function as x approaches -2 from the right. The graph shows that as x approaches -2 from the right, y approaches +1. So the limit is +1.
c) \( \lim_{x \to -2} f(x) \) is the limit of the function as x approaches -2 from both the left and right. Since the limits from the left and right are different, the overall limit does not exist. So the limit is undefined or does not exist.
d) \( \lim_{x \to 3} f(x) \) is the limit of the function as x approaches 3 from either direction. The graph shows that the y-values are approaching a value around +2. So the limit is +2.
e) \( \lim_{x \to +\infty} f(x) \) is the limit of the function as x approaches infinity. The graph shows that as x goes to infinity (to the right), y is approaching a value of -3. So the limit is -3.
