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Problem 64 Hard Difficulty

Evaluate $ \displaystyle \lim_{x \to 2}\frac{\sqrt{6 - x} - 2}{\sqrt{3 - x} - 1} $.

Answer

$\frac{1}{2}$

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Video Transcript

This is problem number sixty four of us to a calculus eighth edition, section two point three. You've only read Element is expert just to however the function Discriminative quantity six minutes. Mine is too directed by the square root of the quantity. Three minutes X minus one. We will first attempt to multiply variety conjugated of a generator, square root of six minutes, X plus two to the top to the bottom and see what this gives us. Excuse us quantity six point six humaneness for divided by any quantity. Square root of negative three musics minus one times skirt of six point six. Plus it too. And then the top produces two six minutes for which is to my name is X. So, uh, we have not met, have made enough progress, as were so many people to directly substitute too into this problem. So our approach would be will be too much pie with it. Country it of the problem. Ah, term here that we began with this denominator and see if that helps the country. It would be a three minute sex in that quantity square root of the quantity plus one divided by a squared to minus six plus one and then more playing these two together should give us three minutes six and then they didn't want to seem to want anything. That one. Ah, that's more the point by this part. Still six minutes X plus two plus two. And that's the denominator. And the new Reiter is this quantity to my sex. What's the point of where this quantity three minutes, six square root of that plus one. Hear what we're in. End up getting this three minus one with just two minus X. As you noticed. This is the same term here in the new Marine er just to cancel. We're lost over with this amount, so we reduce the limit. Two The limited expertise to ah, the square root of three minus X quantity plus one really square root of the quantity. Six. Money, sex plus two. At this point, we can substitute directly three minutes to his one. Describe it in one plus one directed by six minutes to is for the word for prostitute, which gives us two on the numerator for in the Dominator. Answer. Answer either one half for this limit