Use the definition of continuity and the properties of limits to show that the function f(x) = x2 + 5(x - 2)7 is continuous at x = 3.

Answer:

Since f(x) = x^2 + 5(x - 2)^7 is a sum of polynomial terms, it is continuous for all real numbers. Specifically, f(3) = 3^2 + 5(3 - 2)^7 = 9 + 5(1)^7 = 14. Also, lim(x→3) [x^2 + 5(x - 2)^7] = lim(x→3) x^2 + 5 lim(x→3) (x - 2)^7 = 9 + 5 = 14. Because lim(x→3) f(x) = f(3), the function is continuous at x = 3.