Beryllium (\( Be \)) has an atomic number of 4, and its electron configuration is \( 1s^2 2s^2 \). In \( BeCl_2 \), beryllium forms two covalent bonds with two chlorine atoms. To achieve this, the \( 2s \) orbital and one \( 2p \) orbital of beryllium mix to form two \( sp \) hybrid orbitals. These \( sp \) hybrid orbitals are linear and oriented at an angle of \( 180^\circ \), consistent with the linear geometry of \( BeCl_2 \).
The two \( sp \) hybrid orbitals of beryllium overlap with the unpaired electrons of chlorine atoms to form two sigma (\( \sigma \)) bonds. This results in a linear molecular geometry for \( BeCl_2 \).
Final Answer: The type of hybrid orbital present at the beryllium atom in \( BeCl_2 \) is \( \mathbf{sp} \), which corresponds to option \( \mathbf{(1)} \).