Question

Use the definition of continuity and the properties of limits to show that the function is continuous on the given interval. 17. f(x)=x+x−4 ,[4,[infinity]) 18. g(x)= 3x+6 x−1 ,(−[infinity],−2)

Answer 1

To show that the function is continuous on the given interval [4, infinity), we need to demonstrate existence, uniqueness, and limit. The function f(x) = x + x - 4 satisfies all three conditions.

Step-by-step explanation:

To show that a function is continuous, we need to show that it satisfies three conditions: existence, uniqueness, and limit. Let's consider the function f(x) = x + x - 4 on the interval [4, infinity).

Existence: The function is defined for all values of x in the interval [4, infinity).

Uniqueness: The function is a polynomial, so it is continuous everywhere.

Limit: We need to show that the limit as x approaches any value in the interval [4, infinity) exists and equals the function value at that point. Taking the limit as x approaches a value in the interval, we get:

lim(x->a) (x + x - 4) = a + a - 4 = 2a - 4

This limit exists and equals 2a - 4 for any value of a in the interval [4, infinity). Therefore, the function is continuous on the given interval [4, infinity).