Evaluate $\lim_{x\ \to\ \infty}\left(\,\sqrt{\,x^{4} + ax^{2} + 1\,}\, - \,\sqrt{\,x^{4} + bx^{2} +1\,}\,\right)$ 
I rewrote the function to the form
  $$
x^{2}\left(\,
\sqrt{\,1 + \dfrac{a}{x^{2}} + \dfrac{1}{x^{4}}\,}\,-\,
\sqrt{\,1 + \dfrac{b}{x^{2}} + \dfrac{1}{x^{4}}\,}\,\right)
$$
  and figured that the answer would be $0$, but apparently this is wrong.
The correct answer is $\displaystyle{{1 \over 2}\left(\,a - b\,\right)}$. 


Hint:
$$\lim_{x \to \infty} \frac{(\sqrt{x^4 + ax^2 +1} - \sqrt{x^4 + bx^2 +1})(\sqrt{x^4 + ax^2 +1} + \sqrt{x^4 + bx^2 +1})}{\sqrt{x^4 + ax^2 +1} + \sqrt{x^4 + bx^2 +1}}$$
The correct answer is $\frac{1}{2}(a-b)$.
