Question

Lim x to 3 (-x^2)/((x^2)- 6x + 9)

substitution i get -9/0
factoring i get same answer
would that mean the answer is infinity?
if not how could i prove that it isn't infinity

Answer 1

Often if you do not trust your answer, try numbers for example
say x = 3.1
-9.61 / (9.61 - 18.6 + 9)
= - 9.61 / .01 = -961
next try x = 3.01
but you get the idea. I think your answer is correct. The result is -9/0 which is undefined or infinite.

Answer 2

since x^2-6x+9 = (x-3)^2 the denominator -> 0 while the numerator does not. So, you are correct that the limit is undefined.