In-line writing of a function and of a particular value of it

Question

Say I have a simple expression such as $$ y = \frac{x}{\sqrt{x^2 + p}} $$ where $x,y$ are variables, $x> 0$, and $p$ is a parameter.

I also want to highlight in the same line that $y = 1$ for the special choice $p = 0$.

Added If there's room for such nuances in mathematical writing, this is a sort of additional remark, meant to highlight more the role of $p$ than the final effect that $y = 1$. I am not setting this function myself, it comes from prior knowledge. In words, it would sound like 'and look at this special case, by the way'.

So I was tempted to write down something like $$ y = \frac{x}{\sqrt{x^2 + p}} = (p = 0) = 1 $$ Would this writing be understood? What are (other) conventions to do this neatly? How would you actually name this problem?

Answer 1

This would be better:

$$y = \frac{x}{\sqrt{x^2 + p}}\Big|_{p=0} = 1$$ it means the valus of the expression which is restricted by the specific value.

Answer 2

That would not be understood. Instead write something like "we set $f_p(x)=\frac {x}{\sqrt {x^2+p}} $ with $f_0(x)≡1$".