Question

Find the indicated limit, if it exists:

(lim_x to 2 f(x)), where (f(x)= begincases x + 3 & x < 2 3 - x & x geq 2 endcases)

Answer 1

To find the indicated limit as x approaches 2 for the given function f(x), we need to examine the left-hand and right-hand limits separately. As x approaches 2 from the left, f(x) approaches 5, while as x approaches 2 from the right, f(x) approaches 1. Since the left-hand and right-hand limits are not equal, the indicated limit does not exist.

Step-by-step explanation:

To find the indicated limit as x approaches 2 for the given function f(x), we need to examine the left-hand limit (x < 2) and right-hand limit (x ≥ 2) separately.

• For x < 2, f(x) = x + 3. As x approaches 2 from the left, the function f(x) approaches 2 + 3 = 5.
• For x ≥ 2, f(x) = 3 - x. As x approaches 2 from the right, the function f(x) approaches 3 - 2 = 1.

Since the left-hand limit (5) is not equal to the right-hand limit (1), the indicated limit does not exist.