Question

1. use algebraic reasoning to find lim(x→2⁻) (x² + 4)/(x² - 4). check your answer by graphing.

Explanation:

Step1: Factor the denominator

We know that $x^{2}-4=(x + 2)(x - 2)$. So the function becomes $\lim_{x
ightarrow2^{-}}\frac{x^{2}+4}{(x + 2)(x - 2)}$.

Step2: Analyze the left - hand limit

As $x
ightarrow2^{-}$, the numerator $x^{2}+4$ approaches $2^{2}+4=8$. The factor $x + 2$ approaches $2+2 = 4$, and the factor $x - 2$ approaches $0$ from the negative side.
Let $f(x)=\frac{x^{2}+4}{x^{2}-4}=\frac{x^{2}+4}{(x + 2)(x - 2)}$. Then $\lim_{x
ightarrow2^{-}}\frac{x^{2}+4}{(x + 2)(x - 2)}=\lim_{x
ightarrow2^{-}}\frac{x^{2}+4}{x + 2}\cdot\frac{1}{x - 2}$.
$\lim_{x
ightarrow2^{-}}\frac{x^{2}+4}{x + 2}=\frac{8}{4}=2$, and $\lim_{x
ightarrow2^{-}}\frac{1}{x - 2}=-\infty$.
So, $\lim_{x
ightarrow2^{-}}\frac{x^{2}+4}{x^{2}-4}=-\infty$.

Answer:

$-\infty$