Due to some major surgery, I am limited in the use of my left arm and am therefore unable to lift more than 5 lbs with it. Even at 5 lbs, I my range of motion has been reduced drastically. I am currently running and doing various ab workouts. Is there any suggestions as to what else could be done to help get fit with the current limitations of the arms?

weight loss is more of a diet thing than an exercise thing

The best thing to do would be to consult your doctor since like you said, you just went through major surgery. Most likely they'll suggest you to take physical therapy to help regain the strength and motion back into your arm. 

I'd agree with the above but in terms of pure exercise for fat loss without using the arms the best best are probably running and hiit.

 

Hiit being where you do intervals at ~70% of effort with a short break between them, most sites seem to recommend 30 seconds work to 10 seconds rest, but you may need to start off with more rest. You can kinda thrown what you like in to a hiit routine like doing high-knee running on the spot, speed-skater jump things, squats, crunches, push ups, burpees, star jumps w/e you like really as long as it gets you moving. Ideally a hiit routine is around 9-20 minutes long.

 

But it really depends what your end goals are you've mentioned weight loss and 'getting fit' but they kinda cover many things.

Getting fit general means losing fat, gaining muscle and improving stamina (and possibly flexibility) - gaining muscle would be tricky with arm limitations, but fat loss through cardio and pushing stamina through cardio is doable (hiit is good for pushing vo2 max and thus stamina).

Weight loss on the other hand is a problematic term: To get healthier doesn't necessarily mean you just want to lose weight. It's about fat vs muscle; for many people burning fat and putting on muscle can actually increase their weight to begin with but that's not a bad thing esepcially when you consider by measures such as BMI people who don't fit the 'average' body type can easily hit obsese ranking but actually be extremely fit and healthier than some 'average' with perfect bmi.

 

And of course above all else getting your eating right as a basis is important.

I wanted to ask what hiit is, but google told me.

Other than that, you don't need to do complex stuff at all. Best method is hiking in hilly or mountainous terrain. Low intensity endurance sports are good too (cycling at 60% of max heart rate for a few hours). 

Eating healthy speaks for itself, but just what healthy food actually is is not at all straightforward.

weight loss is more of a diet thing than an exercise thing

 

Agreed, I think the statistic is something like 74% diet, 26% exercise.