But then when you do that simplification and factoring out that x minus, or canceling out that x minus four, that gap would disappear. And so that's essentially what you're doing. You're trying to find the limit as we approach that gap, which we got right there. Now this final one, this is dealing with trig identities. And in order to do these, you have to be pretty adept at your trig identities. So if we're saying the limit as, let me do that in a darker color. So if we're saying the limit as x approaches zero of sine of x over sine of two x, well, sine of zero zero, sine of zero zero, you're gonna get zero over zero.