4. For what value of the constant k the function
f(x)= x^ 2 -4/x-2 if x=2,
2k - x if x=2
is continuous at x = 2
A) -3
B) -1
C) 3
D) 1
We need to find the value of k such that the two parts of the function match up at x=2.
First, let's find the limit of the function as x approaches 2.
lim (x->2) f(x) = lim (x->2) (x^2 - 4)/(x - 2)
We can simplify this using factorization:
lim (x->2) f(x) = lim (x->2) [(x+2)(x-2)]/(x-2)
= lim (x->2) (x+2)
= 4
So, for the function to be continuous at x=2, we need f(2) = 2k - 2 = 4.
Therefore, k = 3.
The answer is C) 3.