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The instant at which bulb is switched on, its power is maximum, then it decreases to a steady value.

This can be explained by the increasing resistance due to heating of the filament. (P = V*2*/R). My doubt in the above is in the steady value. Logically, I think as the resistance keeps increasing, the fractional change in it becomes small enough to say that resistance becomes constant, but how do I get to this value mathematically. I tried but could not solve it. My effort : enter image description here

I thought of getting a function of power and time in which at t = infinity, the constant value of power could be obtained. But I am inable to arrive there.Is there a mistake in the approach or the calculation? Please help.

Thanks.

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