What is the output and Ib when transistor beta is infinite?

What happens when transistor beta is infinite? what will be the output of the transistor? Is Ib equal to zero?

When beta gets large, the base current Ib gets small. When the base current is small compared to the other currents in the base circuit (as happens in modern high-speed circuits), it is useful to consider the transistor as a voltage-controlled device: Collector current Ic is exponentially dependent on base-to-emitter voltage Vbe, and transconductance = Ic/Vbe = Gm = Ic/Vt. Vt is the thermal voltage, proportional to absolute temperature. I recall Vt is 26 millivolts at room temperature, but I may be recalling incorrectly.

Correction: transconductance = dIc/dVbe = Gm = Ic/Vt where d indicates derivative, rate-of-change.

https://en.wikipedia.org/wiki/Bipolar_junction_transistor discusses this in detail, and confirms that Vt = 26 mV at T = 300 kelvins. Scroll down to Ebers-Moll model.

Take Darlington transistors as a reference.
Total beta equals
beta1 * beta2.
Therefore the current flowing on the load is equal to
I (load) = i (b) * beta_tot
If the beta is infinite, we have:
- if Ib = 0, Ic = 0 (ib multiplicated by 0 is 0)
- if Ib> 0, Ic = Ib * beta.

The current is limited by:
- maximum current delivered by the load battery
- maximum current supported by the transistor

Giovanni