Final answer:
To redefine f(4) so that x=4 is continuous in the function f(x) = (x-4)/(x^2-16), we find the limit of the function as x approaches 4 and assign that value to f(4). The limit evaluates to 1/8.
Step-by-step explanation:
To redefine f(4) so that x=4 is continuous, we need to find the limit of the function as x approaches 4. Let's evaluate the limit:
limx→4 f(x) = limx→4 (x-4)/(x²-16)
Since 4 is not in the domain of the function, we can rewrite the function as:
f(x) = (x-4)/((x-4)(x+4))
Now, cancelling out the common factor of (x-4), we get:
f(x) = 1/(x+4)
So, to redefine f(4) so that x=4 is continuous, we can assign the value of the function at x=4 as the value of the limit:
f(4) = limx→4 f(x) = 1/(4+4) = 1/8.