determine lim x→2 2x - 4 / x² - 4. if the limit does not exist, enter dne.

Explanation:

Step1: Factor the expressions

First, factor the numerator $2x - 4=2(x - 2)$ and the denominator $x^{2}-4=(x + 2)(x - 2)$. So the function becomes $\lim_{x\rightarrow2}\frac{2(x - 2)}{(x + 2)(x - 2)}$.

Step2: Simplify the function

Cancel out the common - factor $(x - 2)$ (since $x\neq2$ when taking the limit), we get $\lim_{x\rightarrow2}\frac{2}{x + 2}$.

Step3: Substitute the value of $x$

Substitute $x = 2$ into $\frac{2}{x+2}$, we have $\frac{2}{2 + 2}=\frac{2}{4}=\frac{1}{2}$.

Answer:

$\frac{1}{2}$