Limit, x->2, [sqrt{x+2} - sqrt{3x-2}] / [sqrt{5x-1} - sqrt{4x+1}]

kiu1

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40. Evaluate the following:

. . . . .\(\displaystyle \displaystyle \lim_{x \rightarrow 2}\, \dfrac{\sqrt{\strut x\, +\, 2\,}\, -\, \sqrt{\strut 3x\, -\, 2\,}}{\sqrt{\strut 5x\, -\, 1\,}\, -\, \sqrt{\strut 4x\, +\, 1\,}}\)



I have tried to multiply the conjugate over the conjugate (sqrt5x-1 +sqrt 4x+1)/(sqrt5x-1 +sqrt 4x+1),
then the upper number becomes even more complicated as there are so many square root.
 

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40. Evaluate the following:

. . . . .\(\displaystyle \displaystyle \lim_{x \rightarrow 2}\, \dfrac{\sqrt{\strut x\, +\, 2\,}\, -\, \sqrt{\strut 3x\, -\, 2\,}}{\sqrt{\strut 5x\, -\, 1\,}\, -\, \sqrt{\strut 4x\, +\, 1\,}}\)



I have tried to multiply the conjugate over the conjugate (sqrt5x-1 +sqrt 4x+1)/(sqrt5x-1 +sqrt 4x+1),
then the upper number becomes even more complicated as there are so many square root.
You gave up too soon AND you already did too much work trying to multiply that annoying numerator.

Do this:
1) Do NOT multiply the complicated expression in the numerator. Leave it factored.
2) Do the same conjugate treatment for the original numerator. (Really, JUST like you did for the denominator.)
3) Again, DON'T multiply the new mess in the denominator. Leave it factored.
4) Remember your best Rational Function Theory - the parts about vertical asymptotes and holes.

Let's see what you get.
 
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