Question

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

Answer 1

The limit is `-1/3`

Explanation:

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

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

Explanation:

`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`