# f(x) = (x^2 - 4)/(x-2) if x not equal to 2

f(x) = k if x = 2

for which value of k is f continuous at x = 2

f(x) = x +2 (obtained by factoring the numerator and canceling "x-2"

When x=2, f(x) = 4

So make k = 4 at x=2 and you have a continuous function.

## To find the value of k for which f is continuous at x = 2, we need to ensure that the limit of f(x) as x approaches 2 from both sides is equal to the value of f(x) at x = 2.

From the given function, f(x) = (x^2 - 4)/(x - 2) for x ≠ 2, and f(x) = k for x = 2.

To find the limit as x approaches 2, we can substitute the value x = 2 into the function:

lim(x→2) ((x^2 - 4)/(x - 2))

By direct substitution, this limit is undefined since it results in division by zero.

However, if we factor the numerator, we can cancel out the common factor of (x - 2):

lim(x→2) ((x + 2)/(1))

Now, we can calculate the limit by directly substituting x = 2:

lim(x→2) (2 + 2) = 4

To ensure continuity at x = 2, the limit must match the value of f(x) at x = 2. Therefore, we need to make k equal to 4 when x = 2.

So, in order for f to be continuous at x = 2, the value of k should be 4.