$lim_(x rarr 4) (3 - sqrt(5 + x))/(1- sqrt(5 - x)) = ?$

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Answer 1 · 2018-03-08T16:41:53

The limit is $-1/3$

It is similar to both

$lim_(xrarr4)(3-sqrt(5+x))/(x-4)$ and to $lim_(xrarr4)(x-4)/(1-sqrt(5-x))$

but both in one expression.

So multiply $(3-sqrt(5+x))/(1-sqrt(5-x))$ by

$((3+sqrt(5+x)))/((3+sqrt(5+x))) * ((1+sqrt(5-x)))/((1+sqrt(5-x)))$ to get:

$lim_(xrarr4)((9-(5+x))(1+sqrt(5-x)))/((3+sqrt(5+x))(1-(5-x))$

$= lim_(xrarr4)((4-x)(1+sqrt(5-x)))/((3+sqrt(5+x))(-(4-x))$

$= lim_(xrarr4)(-(1+sqrt(5-x)))/(3+sqrt(5+x))$

$= (-(1+sqrt1))/(3+sqrt9) = -2/6 = -1/3$

Answer 2 · 2018-03-08T16:43:04

The limit should approach -1/3, I screwed up the original answer.

$lim_(x rarr 4) (3-sqrt(5+x))/(1-sqrt(5-x))$

first multiply the top and bottom by the conjugate of the numerator and the conjugate of the denominator

$(3-sqrt(5+x))/(1-sqrt(5-x))*$

$(3+sqrt(5+x))/(3+sqrt(5+x))*(1+sqrt(5-x))/(1+sqrt(5-x))$

$= (4-x)/(-(4-x))*(1+sqrt(5-x))/(3+sqrt(5+x))$

$=-(1+sqrt(5-x))/(3+sqrt(5+x))$

plug in the limit value to get your answer:

$=-(1+sqrt(5-4))/(3+sqrt(5+4))$
$=-(1+1)/(3+3) = -2/6 = -1/3$

Tony B

lim_(x rarr 4) (3 - sqrt(5 + x))/(1- sqrt(5 - x)) = ?lim_(x rarr 4) (3 - sqrt(5 + x))/(1- sqrt(5 - x)) = ?