Continuity Answers
Use the definition of continuity and the properties of limits to show that the function is continuous at x = 2.
Use the definition of continuity and the properties of limits to show that is continuous at
Explain why the function is not continuous at
h is not defined at x = −5
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.
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.
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.
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.
Determine whether or not the function is continuous at
continuous
Determine all numbers at which the function is continuous.
continuous at every real number except x = −1
Use the definition of continuity and the properties of limits to show that the function is continuous at x = -2.
Use the definition of continuity and the properties of limits to show that the function is continuous at x = 36.
Use the definition of continuity and the properties of limits to show that the function is continuous at x = -2.
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