12. use algebraic reasoning to find $lim_{x\rightarrow2^{-}}\frac{x^{2}+4}{x^{2}-4}$. check your answer by…

Explanation:

Step1: Factor the denominator

The denominator $x^{2}-4=(x + 2)(x - 2)$. So the function is $\frac{x^{2}+4}{(x + 2)(x - 2)}$.

Step2: Analyze the left - hand limit

As $x\to2^{-}$, we consider values of $x$ that are very close to 2 but less than 2. Let's substitute values close to 2 from the left - hand side into the function $\frac{x^{2}+4}{(x + 2)(x - 2)}$. The numerator $x^{2}+4$ approaches $2^{2}+4=8$ as $x\to2$. The factor $(x + 2)$ approaches $2+2 = 4$ as $x\to2$, and the factor $(x - 2)$ approaches 0 from the negative side as $x\to2^{-}$.

Step3: Calculate the limit

We have $\lim_{x\to2^{-}}\frac{x^{2}+4}{x^{2}-4}=\lim_{x\to2^{-}}\frac{x^{2}+4}{(x + 2)(x - 2)}$. Since the numerator approaches 8, $(x + 2)$ approaches 4, and $(x - 2)$ approaches 0 from the negative side, the limit is $-\infty$.

Answer:

$-\infty$