Financing a 3-4 unit building help!

[Bassim Audi]

I'm new to BP and to the investment real estate world. I'm working on finding my first property to buy. My intention is to find a 3-4 Unit building to live in one of the units and utilize the rest to house hack my mortgage payments. Where I am currently falling short is that a lot of the buildings I am finding cost quite a bit more than what the FHA/conventional in my area (Chicago) allows for and I don't have the capitol to cover the difference. IE a 3-unit building sell price $640,000 FHA max is $560,000 and I have roughly 50k in hand. Am I living a pipe dream or is there something financially I'm missing?

I'm still working my way through all the content BP has to offer but I thought I would try and engage with smarter people while I do. Thanks in advance for you input!