Question

Finally, suppose m1→∞, while m2 remains finite. what value does the the magnitude of the tension approach?

Answer 1

The answer is not infinity. 
Consider FBDs of each mass with the direction of motion of m1 as positive 

m1*g-T=m1*a 

T-m2*g=m2*a 

assuming mass less cord and mass less, friction less pulley 
the accelerations are equal 
a=(T-m2*g)/m2 

m1*g-T=m1*(T-m2*g)/m2 
do some algebra 
m1*g-T=m1*T/m2-m1*g 
2*m1*g=T*(1+m1/m2) 
2*m1*m2*g=T*(m2+m1) 
2*m1*m2*g/(m2+m1)=T 
now take the limit of T as m1->infinity 
T=2*m2*g 

this is intuitively correct since the maximum acceleration of m1 is -g, the cord transfers the acceleration to m2, which is being acted on by gravity downward and an upward acceleration of g. Therefore the maximum acceleration of m1 is 2*g upward.