Scopes are 1 definite area where you get what you pay for, Nikon, Bushnell and Redfield all produce good scopes just avoid their bargain basement models.
I have a 1.5-4.5x26 scope on my .32 special Marlin I can keep 4 for 4 shots on a 8" pie plate at 200 yards with standard 170 grain federal blue box.
My father has a .30-30 with a 2.5x20 leupold ultralight and its also capable of the same accuracy, I just have to concentrate.
I've got a 3-9x40 Bausch & Lomb scope on my .30-06 and have never actually shot big game with it set on 9x, I've made shots out to a lasered 234 yards, (my longest shot on big game) with it set on 3x. I think people generally over scope their guns. Particularly light lovely handling carbines like the marlin.
I'd happily hunt all of North america with a 2.5x scope. In a variable scope I wouldn't go larger then 2-7, 2.5-8 or 3-9x for a general purpose rifle.
In your budget you can usually find deals on high end fixed and lower power variables that's part of the reason for my suggesting a smaller scope, clear optics are more important then magnification when it comes to shooting deer at less then 300 yards. The other is a smaller scope doesn't impact the balance of a light weight carbine as much for carrying and shouldering quickly.