Final answer:
The limit of f(x) as x approaches 8, where f(x) is defined differently for x < 8 and x ≥ 8, is 59.
Step-by-step explanation:
The limit of a function at a specific point can be found by evaluating the function as x approaches that point from both sides. In this case, we need to evaluate f(x) = x² - 4x + 27 as x approaches 8 from the left (x < 8) and f(x) = -x² + 4x - 23 as x approaches 8 from the right (x ≥ 8).
When x < 8, f(x) = x² - 4x + 27. So we substitute x = 8 - h into the function and evaluate the expression as h approaches 0 to find the limit from the left. Similarly, when x ≥ 8, f(x) = -x² + 4x - 23. We substitute x = 8 + h into the function and evaluate the expression as h approaches 0 to find the limit from the right.
Let's perform these calculations step by step:
Left Limit: lim x→8- f(x) = lim h→0 (8 - h)² - 4(8 - h) + 27
= lim h→0 64 - 16h + h² - 32 + 4h + 27
= lim h→0 h² - 12h + 59
= 59
Right Limit: lim x→8+ f(x) = lim h→0 (8 + h)² - 4(8 + h) + 27
= lim h→0 64 + 16h + h² - 32 - 4h + 27
= lim h→0 h² + 12h + 59
= 59
Since the left limit and right limit both converge to the same value of 59, we can conclude that the limit of f(x) as x approaches 8 exists and is equal to 59.