evaluate the limit. if the limit does not exist, enter dne. limx→3 x² - 6x + 9 / x² - x - 6

Explanation:

Step1: Factor the numerator and denominator

The numerator $x^{2}-6x + 9=(x - 3)^{2}$, and the denominator $x^{2}-x - 6=(x-3)(x + 2)$. So the function becomes $\frac{(x - 3)^{2}}{(x-3)(x + 2)}$.

Step2: Simplify the function

Cancel out the common factor $(x - 3)$ (for $x\neq3$), we get $\frac{x - 3}{x+2}$.

Step3: Substitute $x = 3$ into the simplified function

Substitute $x=3$ into $\frac{x - 3}{x+2}$, we have $\frac{3-3}{3 + 2}=\frac{0}{5}=0$.

Answer:

$0$