When I try this:

sage: y = gamma(x+1/2)/gamma(x)/sqrt(x)
sage: limit(y,x=oo)
0
sage: plot(y,x,1,50)

I observe that the limit appears to be 1, not 0. 

Indeed, Mathematica confirms:

In[1]:= y = Gamma[x + 1/2]/(Gamma[x]*Sqrt[x])
Out[1]= Gamma[1/2 + x]/(Sqrt[x] Gamma[x])

In[2]:= Limit[y, x -> Infinity]
Out[2]= 1

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