Evaluate the given limit:

Question

lim x->3 ((x-3)/(sqrt(x) - sqrt(6-x)))

 

No l'hospital, use derivative limit rules.

Answer 1

Multiply numerator and denominator by the conjugate of the denominator

(x-3)/(√x -√(6-x)) = (x-3)(√x + √(6-x)/(x -6+x) = (x-3)(√x + √(6-x))/(2(x-3))

= (√x + √(6-x))/2

lim x→3 (√x + √(6-x))/2 = (√3 + √3)/2 = √3