The rational function f(x) has a hole at x = 3 and horizontal asymptotes at y = -2. As x approaches 3 from the left, f(x) approaches positive infinity, and as x approaches 3 from the right, f(x) approaches negative infinity. As x approaches positive or negative infinity, f(x) approaches -2.
Local Behavior
As x approaches 3 from the left (3-), f(x) approaches positive infinity (∞).
As x approaches 3 from the right (3+), f(x) approaches negative infinity (-∞).
End Behavior
As x approaches positive infinity (∞), f(x) approaches -2.
As x approaches negative infinity (-∞), f(x) approaches 2.
Therefore, the following local and end behaviors are correct:
As x approaches 3 from the left (3-), f(x) approaches positive infinity (∞).
As x approaches 3 from the right (3+), f(x) approaches negative infinity (-∞).
As x approaches positive infinity (∞), f(x) approaches -2.
As x approaches negative infinity (-∞), f(x) approaches 2.
The graph of the rational function f(x) has a vertical asymptote at x = 3, with the graph approaching positive infinity as x approaches 3 from the left and negative infinity as x approaches 3 from the right. This indicates that f(x) has a hole at x = 3. As x approaches positive or negative infinity, the graph of f(x) approaches the horizontal asymptote y = -2.
Therefore, the following local and end behaviors are correct:
As x approaches 3 from the left (3-), f(x) approaches positive infinity (∞).
As x approaches 3 from the right (3+), f(x) approaches negative infinity (-∞).
As x approaches positive infinity (∞), f(x) approaches -2.
As x approaches negative infinity (-∞), f(x) approaches 2.