I think you dropped a 3 in your calculation (or in your typing of your answer). The equation of the tangent line is:
Now as to the asymptotes: Remember, , so your hyperbola in standard form is and so the asymptotes are .
Alternatively, the asymptotes for such a hyperbola are of the form where m= . Parametrically, and for x to go to infinity, t must approach . So .