Simplify this $1/Sqrt(1+1/t)$ so that it works for t=0?

Questions
1

How would I get Mathematica to simplify 1/Sqrt(1+1/t) so that it works for t = 0? I tried the FullSimplify and Simplify commands and am not sure what else to do, even after taking a look at the built in symbol’s information.

2

I am not sure what you expect as a result. Here is a transformation using a replacement rule:

1/Sqrt[1 + 1/t] /. (1/Sqrt[a_] :> Sqrt[Together[1/a]])

You may also try PowerExpand, which is risky:

Simplify[PowerExpand@Together[1/Sqrt[1 + 1/t]],
 t < 0 || t > 0]

The result does not seem particularly “simpler” than the original expression.

3

Well, the sort of answer I’m looking for is just obtaining the form |t|/Sqrt[t^2+t] (or something similar) which will give 0 when zero is plugged in, something that doesn’t return “1/0” errors.

4

The expression |t|/Sqrt[t^2+t] gives errors too when t=0. Try with this:

Sqrt[t/(1 + t)]
5

You can get the expression that Gianluca gave you with

Simplify[Apart[1/Sqrt[1 + 1/t]], Assumptions -> t >= 0]

Evidently you need the Apart because MMA thinks that the original expression is simple enough so you can force an expansion and re-simplify with a positive assumption.

Regards

Neil

6

One more way:

Simplify[1/Sqrt[1 + 1/t] // ComplexExpand, {t >= 0}]
(* Sqrt[t/(1 + t)] *)