I have a function $\frac{x^{1-\gamma}-(xy)^{\frac{1-\gamma}{2}}}{y - x}$ that has a singularity at $y=x$ but the limit as $x \to y$ is well defined, namely, $\lim_{x \to y}\frac{x^{1-\gamma}-(xy)^{\frac{1-\gamma}{2}}}{y - x} = \frac{1}{2}(\gamma - 1)y^{-\gamma}$. My goal is to re-express the function to avoid this singularity and I would like to know if there exists a theorem that guarantees the existence of such a function. Also, is there a general technique for tackling such a problem or is it more based on experience and trial and error? $x$, $y$, and $\gamma$ are all real, positive values.
As an example, I have a similar function, $\frac{y - (xy)^{\frac{1}{2}}}{y - x}$ that I can re-express as $\frac{y + (xy)^{\frac{1}{2}}}{x + y + 2(xy)^{\frac{1}{2}}}$ to avoid the singularity.
If you mean to find a “closed” expression, then you can't, in general. Consider the function $$ f(x)=\begin{cases} \dfrac{\sin x}{x} & x\ne 0 \\[4px] 1 & x=0 \end{cases} $$ This is not expressible under a “single formula” and your case is very similar. Indeed, you have $$ f(x)=\frac{x^{1-\gamma}-(xy)^{\frac{1-\gamma}{2}}}{y - x} $$ The limit for $x\to y$ is the derivative at $y$ of $g(x)=(xy)^{(1-\gamma)/2}-x^{1-\gamma}$.
If you set $\delta=(1-\gamma)/2$, you have $$ g(x)=(xy)^{\delta}-x^{2\delta} $$ For rational $\delta$ you can, in a finite time, find an expression of the form $g(x)=(x-y)q(x)$, for a quite complicated $q$, complexity depending on the least positive denominator needed to express $\delta$. But for irrational $\delta$ this is not possible: such irrational will have rational numbers in every neighborhood and the formula will be different for any of them, but they should transform “continuously” into one another (one can make precise what “continuous” means).