My lease will be up for renewal early next year. I am currently paying $5 less than the high end range for the apartment. Could be rent be legally increased beyond $5.
Unless you're in a public housing unit or government subsidized unit, there is no legal limit to the amount a landlord can charge. Typically, they'll charge what the market will bear.
If they offer you a renewal at a rate you're unconfortable with, you're under no obligation to accept. On the other hand, they're under no obligation to offer you a renewal.
This answer is intended as informational only, and does not constitute legal advice or form an attorney-client relationship between us.
I agree, additionally keep in mind that you have no obligation to renew, just like the landlord has no obligation to keep the rent at a certain price or renew the lease. I encourage you to treat this practically, and consider all factors such as moving costs, etc. Good Luck.