Maximum Value for European Puts and Calls

CFA Level I
#1

They say the maximum value for a European put option would be the present value of the strike price. However shouldn’t the maximum value for a put option technically be infinity, because that will entail the greatest chance for you to make money on your investment (if you were long a put option).

Also, I do not get why the minimum value for a call option would be the underlying price of the stock itself. But shouldn’t investors see the max value in the call option when the strike price is the lowest possible (for the greatest chance of exercising and making a profit between the underlying stock in the minimal strike price?)

I may be overthinking this, but it is hard for me to just remember things for the sake of it. I like to know things intuitevly (for the better for worse).

#2

For a put, the least a share can go is to ‘0’. So going long a put, the maximum payoff is Strike minus 0 and the PV is nothing but the PV of the strike. For a call, the minimum value is the stock price itself since if you’re long a call and the stock price is below the strike, you can buy the stock itself at its price.