Question

Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a. f(x) = x⁴ - 4x³ + 5, a = -1?

Answer 1

To show that the function f(x) = x⁴ - 4x³ + 5 is continuous at a = -1, we need to check three conditions: 1) The function is defined at a = -1. 2) The limit of the function as x approaches a = -1 exists. 3) The limit of the function as x approaches a = -1 is equal to the value of the function at a = -1.

Step-by-step explanation:

To show that the function f(x) = x⁴ - 4x³ + 5 is continuous at a = -1, we need to check three conditions: 1) The function is defined at a = -1. Since substituting x = -1 into the function gives f(-1) = (-1)⁴ - 4(-1)³ + 5 = 11, the function is defined at a = -1. 2) The limit of the function as x approaches a = -1 exists. We can find this limit by evaluating the function as x approaches -1. lim(x->-1) [(x⁴ - 4x³ + 5)] = (-1)⁴ - 4(-1)³ + 5 = 11. Therefore, the limit exists. 3) The limit of the function as x approaches a = -1 is equal to the value of the function at a = -1. We have already determined that the limit as x approaches -1 is 11, and we also know that f(-1) = 11. Since the limit and the value of the function at a = -1 are the same, the function is continuous at a = -1.