Calculate the limit at x=0 Find the limit of $f(x)=\frac{\sqrt{a^2-ax+x^2}-\sqrt{a^2+ax+x^2}}{\sqrt{a-x}-\sqrt{a+x}}$ (at x=0) so that $f(x)$ becomes continuous for all $x$. My answer is $2\sqrt{a}$. Am I right?
Sorry to state the question incorrectly. We have to define $f(x)$ at $x=0$ such that $f(x)$ is continuous for all $x$. In that case my answer is $2\sqrt{a}$.
Let f(x)=$[x]$+$[-x]$ be a function where [.] stands for the greatest integer not greater than x. For any integer $m$, what can we say about $lim_{x \to m}$. Is $f(x)$ contiuous at $x=m$. Sorry for asking such vague question but I am forgetting the exact wording and the options given. This was a question asked in a class test.

As $x\to 0$ you have 
$$\sqrt{a^2 - ax + x^2} - \sqrt{a^2 + ax + x^2}= {-2ax\over \sqrt{a^2 - ax + x^2} + \sqrt{a^2 + ax + x^2}}\sim {-2ax\over 2a} = -x $$
and
$${1\over \sqrt{a-x}-\sqrt{a+x}} = {-2x\over  \sqrt{a-x}+\sqrt{a+x}}\sim -{x\over \sqrt{a}} $$
Now divide to see tht you get $\sqrt{a}$ for the limit of the ratio.
