What is the limit of x approaching 2 where the sqrt of bracket 6 minus x bracket minus 2 all divided by sqrt of bracket 3 minus x bracket minus 1?

Answer 1

This is a case where we can apply l'Hopital's rule, separately calculating the derivatives of the numerator and denominator, then take the limit using those:

Limx→2 [(6 - x)1/2 - 2] / [(3 - x)1/2 - 1]

= Limx→2 [1/2(6 - x)-1/2 (-1)] / [1/2(3 - x)-1/2 (-1)]

= Limx→2 (3 - x)1/2 / (6 - x)1/2

= 1/2