Question

How do you find the limit of `(1/sqrtx-1/sqrt2)/(x-2)` as `x->2`?

Answer 1

Begin by rewriting it without the difference on fractions in the numerator of a fraction. Then see where that leads.

Explanation:

`(1/sqrtx-1/sqrt2)/(x-2) = ((1/sqrtx-1/sqrt2))/((x-2)) * sqrt(2x)/sqrt(2x)`

`= (sqrt2-sqrtx)/(sqrt(2x)(x-2))`

`= (-(sqrtx-sqrt2))/(sqrt(2x)(sqrtx-sqrt2)(sqrtx+sqrt2))`

`= (-1)/(sqrt(2x)(sqrtx+sqrt2))`

So,

`lim_(xrarr2) (1/sqrtx-1/sqrt2)/(x-2) =lim_(xrarr2) (-1)/(sqrt(2x)(sqrtx+sqrt2))`

`= (-1)/(sqrt4(sqrt2+sqrt2)) = (-1)/(4sqrt2) = - sqrt2/8`