plot and limit are not in agreement

I am plotting the following function $ \frac{\sqrt{t \text{$ \omega_c $ }}-\sqrt{\pi } e^{\frac{1}{t \text{$ \omega_c $ }}} \text{erfc}\left(\frac{1}{\sqrt{t \text{$ \omega_c$ }}}\right)}{(t \text{$ \omega_c$ })^{3/2}}$ for $ \omega_c=1$

 (Sqrt[t*\[Omega]c] - E^(1/(t*\[Omega]c))*Sqrt[Pi]*Erfc[1/Sqrt[t*\[Omega]c]])/(t*\[Omega]c)^(3/2) 

and I am obtaining

enter image description here

where we clearly see that the function diverges as $ t \rightarrow 0$ . What is interesting is that if I perform the limit through Mathematica for $ t\rightarrow 0$ I obtain

enter image description here

which clearly disagrees with the graph. It is important to note that for $ t\approx 10^{-3}$ the graph tends to the same correct limit, indicated by the blue line. Is there any way to increase the precision of the plot?