Final answer:
To determine the limits of the piecewise function g(x), one must consider the behavior from both sides of the target x-value. For x > 0, the limit is the natural logarithm of x. As x approaches 0 from the right, the limit is negative infinity, while the left limit is undefined, and the value at x = 0 is defined as 2.
Step-by-step explanation:
To determine the limits of the piecewise function g(x), given by g(x) = ln(x) for x > 0 and g(x) = 2 for x ≤ 0, we must examine the behavior of the function as x approaches a point from both the left and the right side. When x approaches any point greater than 0, the limit is simply the natural logarithm of that point, due to the continuity of the ln(x) function for x > 0.
However, as x approaches 0 from the right, the limit of g(x) = ln(x) would tend towards negative infinity. When approaching from the left, x cannot be negative, so the limit doesn't exist in this context because ln(x) is undefined for x ≤ 0. As for the value at x = 0, g(x) is defined to be 2. In summary, the left limit as x approaches 0 is not defined, the right limit is negative infinity, and the function value at x = 0 is 2.