Evaluating $\lim_{x \to -\infty} \frac{1}{-\sqrt{\frac{1}{x^6}}\sqrt{x^6+4}}$ Problem:

$$\lim_{x \to -\infty} \frac{1}{-\sqrt{\frac{1}{x^6}}\sqrt{x^6+4}}$$

$$\lim_{x \to -\infty} \sqrt{\frac{1}{x^6}}=0$$ so...
$$\lim_{x \to -\infty} \frac{1}{-\sqrt{\frac{1}{x^6}}\sqrt{x^6+4}}=\frac{1}{0}$$
The answer is $-1$ and I know how to get that answer. Where is the mistake in this method though?

Just as $\sqrt{1/x^6}$ goes to $0$, so does $\sqrt{x^6+4}$ go to $\infty$. You cannot substitute just one of these radicals and then simplify, and their unsimplified product is the indeterminate form $0\cdot\infty$ and so cannot be handled directly.
