The limit of the radical expression limx→4 (sqrt(5-x)-1) / (2-sqrt(x)) can be solved without L'Hospital's Rule by rationalizing the numerator and denominator. The key technique involves multiplying both the numerator and denominator by the conjugate of the denominator, 2 + sqrt(x), and subsequently applying the conjugate of the numerator, sqrt(5-x) + 1. This method simplifies the expression, allowing for cancellation of terms and ultimately leading to the correct limit evaluation.
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No. I think what HallsofIvy is trying to tell you is that you have to multiply by the conjugate twice. See my post in this thread.phrox said:The only thing I see you can do with that is square it I guess?I'm not sure... Subbing in 4 will just make it 0.
You are not posting your working, so I have to guess. It sounds like you went through the full multiplication top and bottom each time. Don't do that. Multiply top and bottom by that expression which removes the radical from the denominator, but in the numerator just leave it as a product of two radical expressions. don't multiply that out. Now do the same for the radical expression which is the original numerator - multiply top and bottom by its radical 'conjugate'. You should now have an expression which is a product of two terms in the numerator (one being the conjugate of the original denominator, and the other not involving a radical) and similarly two terms in the denominator.phrox said:By rationalize do you mean multiply the conjugate? I don't see how you can rationalize (multiply top and bottom by the root), but you say you only do it to the top and then only do it to the bottom. I don't think you can do that. Also if I multiply the top conjugate and then bottom conjugate it gets REALLY messy and it can't be right.
(This is all after reading your post)