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 $\displaystyle{{1 \over 2}\left(\,a - b\,\right)}$. 


Substitute $x=1/t,$ then our expression becomes $$\lim_{t \to 0} \frac{\sqrt{1+at^2+t^4}-\sqrt{1+bt^2+t^4}}{t^2}$$
Note that here we can apply the L'Hospitals rule. 
$$\frac{\frac{1}{2}(1+at^2+t^4)^{-1/2}(2at+4t^3)}{2t} \to \frac {a}{2}$$
as $t \to 0.$
Therefore it easy to see that required limit is $$\frac{a-b}{2}.$$
