Question

lim x->4 [sqrt(5-x) -1]/[2 - sqrt(x)]

Answer 1

rationalize the denominator.

Answer 2

rationalizing will still leave the denominator as 0 at the limit.

Answer 3

Sometimes the limit approaches 0, without any manipulation the limit cannot be evaluated because it would result in an undefined value (which is not 0, but undefined)

Answer 4

It has been really long time that I tackled limit. I know I can use the L'Hopital's rule and that makes life very easy but I have to teach someone who has not yet reached that topic in his class yet. I was looking for help year to crack it algebraically.

Answer 5

The limit is just what the function approaches. The limit evaluating to 0 is not "wrong". You just need to check that if you approach the limit from the left and the right, that both limits agree.

Answer 6

the algebra is a pain in the neck
you have to multiply top and bottom by the conjugate of both

Answer 7

\[\frac{(\sqrt{x-4}-1)(\sqrt{x-4}+1)(2+\sqrt{x})}{(2-\sqrt{x})(2+\sqrt{x})(\sqrt{x-4}+1)}\]

Answer 8

it is even a pain just to write it
but now it should be easy enough

Answer 9

what is worse is that i wrote it incorrectly
it should be \[\large \frac{(\sqrt{x-5}-1)(\sqrt{x-5}+1)(2+\sqrt{x})}{(2-\sqrt{x})(2+\sqrt{x})(\sqrt{x-5}+1)}\]

Answer 10

@satellite73 why dont you just multiply num and den by (2-sqrt(x))

Answer 11

\[\sqrt{x-5}-1)(\sqrt{x-5}+1)=x-5-1=x-4\] and
\[(2-\sqrt{x})(2+\sqrt{x})=4-x\]

Answer 12

i don't know, does that work?

Answer 13

no, it wont work, because you have nothing to cancel if you do it

Answer 14

actually it did - satellite63. It is actually sqrt(5 - x) -1 so, it becomes 5 - x -1 i.e. 4 -x. Now we get this term in both numerator and denominator so they cancel each other. Plugging the limit yields 2.

Answer 15

now since
\[\frac{x-4}{4-x}=-1\] you have cancelled out the zero top and bottom and are left with
\[\frac{2+\sqrt{x}}{\sqrt{x-5}+1}\] and you can replace \(x\) by \(4\) at this step

Answer 16

i think we are both off by a minus sign

Answer 17

I think you are mistyping sqrt(5-x) as sqrt(x-5) ;)

Answer 18

\[-\frac{2+\sqrt{x}}{\sqrt{x-5}+1}\]

Answer 19

yeah i mistyped a bunch of stuff'
i am sure you can fix it
the gimmick is to multiply by the conjugate of both and cancel

Answer 20

it really works - i was just looking for the approach and your help was timely.

Answer 21

cheers http://www.wolframalpha.com/input/?i=lim+x-%3E4+%28sqrt%285-x%29+-+1%29%2F%282-sqrt%28x%29%29