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.

EDIT::
$$ \frac{\sqrt{a^2}-\sqrt{a^2}}{\sqrt a - \sqrt a} = \frac{0}{0}\neq 2\sqrt a $$
You have to multiply by conjugate of both terms (in numerator and denominator) and get the following.
$$ \frac{\sqrt{a^2-ax+x^2}-\sqrt{a^2+ax+x^2}}{\sqrt{a-x}-\sqrt{a+x}} \\
= \frac{(a^2-ax+x^2)-(a^2+ax+x^2)}{(a-x)-(a+x)} \times \frac{\sqrt{a-x}+\sqrt{a+x}}{\sqrt{a^2-ax+x^2}+\sqrt{a^2+ax+x^2}} \\
= \frac{-2ax}{-2x}\times \frac{\sqrt{a-x}+\sqrt{a+x}}{\sqrt{a^2-ax+x^2}+\sqrt{a^2+ax+x^2}} 
\\
\text{ taking limit x } \rightarrow 0 \text{ we get}
=a \frac{2\sqrt a}{2 a} = \sqrt a$$
