What is the magnitude of the impulse function at t=0? The unit impulse function, in a few textbooks that I have referred, has a value of 0 at t≠0 , and an area of unity (1). The height of the impulse function also tends to infinity at t=0. But since it's height cannot be it's magnitude (as we generally use it's unit area in calculations), what would be the value/amplitude/magnitude of the unit impulse function at t=0?

Before you say that my question has answered itself, and that it's value is one(1) at t=0, note that this is not defined as such in any standard textbooks. This question has been bothering me for quite a while now, so any attempt at elaborating a solution will be highly appreciated.
The "impulse function" is not actually a function but a distribution, and its value at $t=0$ is not defined.

The impulse only really makes sense when multiplied by a signal inside an integral, where it is used to "select" the value of the signal at specific time instants.

Personally, I try to avoid impulses whenever possible. Look for Lapidoth's book on digital communications (available for free on his website) for a complete treatment of Fourier analysis (as relevant for communications) without a single delta. Not many people share my squimishness with deltas, though, and if you're taking a signals course they are pretty much unavoidable.

Note also that, once you go to the discrete time domain, everything becomes nicer: instead of the Dirac delta, you use the Kronecker delta, which is a regular function and has a value of 1 at $n=0$.