# How to apply equivalences in the Limit?

Actually, I have the following expression $a1$:

ClearAll["Global*"];
$Assumptions = {c > 0, h > 0}; a1 = (c + Sqrt[c^2 + 4 h^2])/(4 h Sqrt[c^2 + h^2]) // FullSimplify  Both$c$and$h$are positive real numbers. However, I want to show by Mathematica that once$c>>h$, i.e.,$c$is much more than$h$, then$a1$could be simplified in the limit case to the following expression which is easier to handle: a2 = 1/(2 h) (1 + h^2/(2 c^2)) // FullSimplify  But I cannot incorporate the equivalency$c>>h$into the command Limit[]. I will be thankful if someone puts some comments to overcome it. • Simplify[ Limit[ Simplify[(c + Sqrt[c^2 + 4 h^2])/(4 h Sqrt[c^2 + h^2]) /. c -> h/z], z -> 0], h > 0] – Artes May 16 '17 at 10:04 • Or Simplify[#, h > 0] & //@ Limit[(c + Sqrt[c^2 + 4 h^2])/(4 h Sqrt[c^2 + h^2]) /. c -> h/z, z -> 0] – Artes May 16 '17 at 10:16 • I think there is a misunderstanding. We should obtain$a2$by imposing the condition$c>>h$on$a1$. Is there a way to reach$a2$from$a1\$? – Fazlollah May 16 '17 at 10:49
• You can expand in a "small parameter" to desired order: Normal@Series[a1 /. h -> \[Epsilon] c, {\[Epsilon], 0, 2}] /. \[Epsilon] -> h/c // FullSimplify`. – jkuczm May 16 '17 at 11:09