I just wanted to know if I make my character a Warrior Mage(one that wears heavy armor most of the time, and casts spells sometimes and uses swords/other melee weapons and a bow) will spreading out my skills too much limit my power or will I still be a force to be reckoned with? Thank you in advance.
There are no classes in Skyrim, and skills level up by being used.
So yes, you can use both combat and magical skills. You do however have a limited number of perks to work with, so you'd be best served by choosing to focus on one or two schools of magic and just one combat style.
You should be able to go deeply into 3-5 perk trees without trouble and with a few points for some others as well.