It sounds like you might be using the wrong filter type if you want to have the filter update with each day. Have you tried applying a relative date filter instead? Quick Start: Relative Date Filters
You would just apply a relative date filter for last 8 days.
There are also ways to approach this through parameters and calculations as well: Creating a Relative Date Filter with Slider | Tableau Software
Not that I want to restrict range. I just want to restrict by default range. By default it should be for 8 days but user should always have liberty to drag it up to last available date in database. and first date should always be today.
They should be able to adjust the range if you expose the relative date filter as a quick filter on the dashboard. The design of the filter may change, but this is in part to help users adjust data as a moving range whereas date range filters are fixed based on a hard-coded start and end date.
If that ideas does not work, you could also look into the parameter approach i linked above.