Using the Jurisdictions Web Map link
Here's how you can find out in which jurisdiction you reside:
Navigate to the City of Austin Jurisdictions Web Map: https://maps.austintexas.gov/GIS/JurisdictionsWebMap/
Click I want to… and then click Find My Jurisdiction
Enter your address (house number and street name only).
From the dropdown list that's displayed, click on the address you're searching for and click Submit.
If the search displays results, look at the Address Description section to see the jurisdiction for the address entered.
The Austin Full Purpose jurisdiction description indicates you qualify for a Resident Library Card free of charge.
What if my address has another jurisdiction description displayed?
If you see another description, such as Austin 2 Mile ETJ, Austin LTD, (another city) Full Purpose, or No Address Found, the address is not within the city limits of Austin and will require a non-resident fee in order to get or renew a Library Card.
Why aren't Austin 2 Mile ETJ and Austin LTD jurisdictions within the city limits of Austin?
The Planning and Zoning Department's FAQ regarding ETJ and Limited Purpose jurisdictions can be found here.