Basically 3Vt (≈80mV) is an input differential voltage range of a BJT diff pair (you can check math analysis in Grey, Hurst, Meyer book "Analysis and design...").
If Vin_diff is higher than this 3Vt value one guy is cut-off, while second is saturated.
However, If your Re is 2.5kΩ, Rc is 20kΩ, input common mode is on ground, VEE is -5.7V and VCC is 12V, then your amplifier is in trouble.
Only left guy might be in active-linear mode, right is always squeezed by large Rc resistor.
So, for input voltage below ground, output is going to be ca Vcc-(Vcc-Vee-Vcesat)Rc/(Rc+Re), while in opposite situation, output is pulled up by Rc to Vcc.
Also, your question is not fully clear. Attached scope screenshot is in time domain and I believe it is showing full range response.
So, asymptotes of what you need?