# What happens when an expression for a physical quantity becomes undefined?

Now I am taking a quite a specific example here. Let's consider a circular race track banked at an angle of $$\theta$$. The coefficient of friction between the track and the tire is $$\mu$$. What's the maximum speed that the car can achieve without skidding out?

Now to solve this problem we draw the car's free body diagram, consider centripetal acceleration and after a lot of math we come up with the expression of $$v_{\rm max} = \sqrt\frac{g\,r\,(\tan \theta + \mu)}{(1-\mu\,\tan\theta)}$$

Now the my question is that what happens when $$\mu\,\tan \theta > 1$$ Obviously, the expression for $$v_{\rm max}$$ becomes undefined but I would like to know that in physical world, what would be the fallout of this?

So answering it mathematically, a good way to think about this is to see what happens when we vary $$\theta$$, as this is the free parameter. I do just want to correct one thing you said: the expression for $$v_{max}$$ actually becomes undefined for $$\mu\;tan\,\theta \geq 1$$.
Anyway, Let's say $$\mu = 0.5$$. Start with $$\theta = 0$$ and increase the $$\theta$$ gradually, which corresponds increasing the incline. As you do, the max speed you can go before skidding out increases. This makes sense, a car would struggle to move up the incline. The speed required to skid out increases asymptotically with $$\theta$$ and at some incline, $$\theta=63.43^\circ, \mu\;tan\,\theta=1$$ and there is no finite speed high enough to skid out.