Find the limit (if it exists). (If an answer does not exist, enter DNE.)

lim f(x) as x approaches 3, where f(x)=
x^2-8x+9 if x<3
-x^2+8x-21 if x>/3 (greater than or equal to 3)

To find the limit of a function as x approaches a specific value, we need to examine the behavior of the function on both sides of that value.

In this case, as x approaches 3 from the left (x < 3), the given function is f(x) = x^2 - 8x + 9. We can calculate the value of the function when x is very close to 3 from the left side. Let's substitute x = 3 - h, where h is a very small positive number approaching 0.

f(3 - h) = (3 - h)^2 - 8(3 - h) + 9
= 9 - 6h + h^2 - 24 + 8h + 9
= h^2 + 2h - 6

Now, as x approaches 3 from the right (x > 3), the given function is f(x) = -x^2 + 8x - 21. Similarly, we can calculate the value of the function when x is very close to 3 from the right side. Let's substitute x = 3 + k, where k is a very small positive number approaching 0.

f(3 + k) = -(3 + k)^2 + 8(3 + k) - 21
= -9 - 6k - k^2 + 24 + 8k - 21
= -k^2 + 2k - 6

In order to find the limit of f(x) as x approaches 3, we need to examine the behavior of these two expressions as h and k approach 0.

Taking the limit as h approaches 0 from the left side:
lim(h->0-) (h^2 + 2h - 6)
= 0^2 + 2(0) - 6
= -6

Taking the limit as k approaches 0 from the right side:
lim(k->0+) (-k^2 + 2k - 6)
= -(0)^2 + 2(0) - 6
= -6

Since the limit from both sides is the same (-6), the limit of f(x) as x approaches 3 exists and it is -6. Therefore, the answer to the question is -6.