Question

1. consider $f(x) = \frac{1}{x^2 - 2}$. evaluate $\lim\limits_{x\to(\sqrt{2})^-} f(x)$. a. 0 b. 1/2 c. $\infty$ d. $-\infty$ e. answer not given

Explanation:

Step1: Analyze left-hand input values

$x \to (\sqrt{2})^-$ means $x < \sqrt{2}$, so $x^2 < 2$.

Step2: Evaluate denominator sign

$x^2 - 2 < 0$, and $x^2 - 2 \to 0$ as $x \to \sqrt{2}$.

Step3: Evaluate the limit

$\lim_{x \to (\sqrt{2})^-} \frac{1}{x^2 - 2} = -\infty$, since we divide 1 by a negative number approaching 0.

Answer:

D. $-\infty$