Solve: should not return empty solutions

Consider the equation $$\frac{-1+\sqrt{1+x^2}}{x}=0$$ which should have a solution $$x=0$$. This is because $$\frac{-1+\sqrt{1+x^2}}{x}=\frac{x^2}{x(1+\sqrt{1+x^2})}=\frac{x}{1+\sqrt{1+x^2}}$$ However, using Solve, Mathematica returns empty solution. Is there any way to let the Mathematica return the correct result? (The code is attached here.)

Solve[(-1 + Sqrt[1 + x^2])/x == 0, x]

• Disable the automatic verifier: In[55]:= Solve[(-1 + Sqrt[1 + x^2])/x == 0, x, VerifySolutions -> False] Out[55]= {{x -> 0}} In general, such solutions should be tested using e.g. Limit. Jan 29, 2019 at 16:42
• @ DanielLichtblau, thanks! Jan 29, 2019 at 17:03

For (-1 + Sqrt[1 + x^2])/x to be zero, its numerator must be zero, so
Solve[(-1 + Sqrt[1 + x^2]) == 0, x]

But, in that case, we have 0/0 for the full formula, so x->0 is not a solution. There are no solutions.