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?

$\frac 1 {-\sqrt {\frac 1 {x^{6}}} \sqrt {x^{6}+4}}$ is nothing but $\frac 1 {- \sqrt {\frac 4 {x^{6}}+1}}$ so the limit is $\frac 1 {-1}=-1$.
