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

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.