This is what I came up with

--SAMPLE DATA

IF OBJECT_ID('tempdb..#table_A') IS NOT NULL

DROP TABLE #table_A;

IF OBJECT_ID('tempdb..#table_B') IS NOT NULL

DROP TABLE #table_B;

CREATE TABLE #table_A (name varchar(20), [address] varchar(20));

CREATE TABLE #table_B (name varchar(20), [address] varchar(20));

INSERT #table_A VALUES ('Tai','#'),('Mike','#');

INSERT #table_B VALUES

('Tai','123 Main St'),('Tai','897 Main St'),('Tai','123 Main St'),

('Mike','456 Broadway'),('Mike','456 Broadway');

--The query

WITH distinct_folks(name, r, [address]) AS

(SELECT DISTINCTname, RANK() OVER (PARTITION BY name ORDER BY [address]),

[address]

FROM #table_B)

SELECT b.name, b.[address]

FROM distinct_folks AS b

JOIN #table_A AS a ON b.name=a.name

WHERE r=1