Is it the eradication of the virus (assuming rhinovirus) by white blood cells? Or does your body somehow adapt to presence of the virus?
For the rhinovirus in particular, it is eradicated.
However for some of the retroviruses and other virus that contain an enzyme called 'integrase', they actually integrate their DNA with yours. That way as your cells go about their business, they make the virus genes too. However, you have some white blood cells called CD8 T Cells that mark these 'compromised cells' for death.
Some viruses though can remain cleverly hidden and don't turn on their own genes in your genome until the immune response goes down, or you stop taking anti-retroviral drugs. HIV is a great example of a virus you can also 'live' with but never actually eradicate. The medicines you take can kill all the circulating virus down to a non detectable level. However, when you stop taking your meds, the DNA integrated in your genome will start making the virus again, thus you will always 'live' and can never eradicate it.