Use the definition of continuity and the properties of limits to show that the function g(x) = (x2 - 3x - 10)(x2 + 2x2 - 5x + 4) is continuous at x = -1.

Answer:

The function g(x) = (x^2 - 3x - 10)(x^2 + 2x^2 - 5x + 4) is a product of polynomials, so it is continuous for all real numbers. Combine terms in the second factor: x^2 + 2x^2 - 5x + 4 = 3x^2 - 5x + 4. Then g(-1) = [(-1)^2 - 3(-1) - 10][3(-1)^2 - 5(-1) + 4] = (1 + 3 - 10)(3 + 5 + 4) = (-6)(12) = -72. By limit laws, lim(x→-1) g(x) = [lim(x→-1)(x^2 - 3x - 10)][lim(x→-1)(3x^2 - 5x + 4)] = (-6)(12) = -72. Since lim(x→-1) g(x) = g(-1), the function is continuous at x = -1.