how to prove that after specific value of negative power, any two exponential functions will be equal?

Question

I need to show that for any two functions $1-e^{-(a+b)}$ and $1-e^{-(a+c)}$ they are approximately equal after a certain limit of $a, 1000000 for example,$, given that the $ b,c \ge 0$ . Is it possible to make approximate proof? means to show that after some value of 'a', the two exponential functions are approximately equal. And any addition to the power for example value 'b' will not play important role in a function. For example here, if a=1000000 the value of the second expression will be very closed to zero.