Explain how lim x->1 ((x^2)-1)/(sqrt(2x+2)-2) = 4

Answer 1

Rationalize the denominator, that is, multiply top and bottom by √(2x+2) + 2 to give you

lim [(x^2 - 1)/(√(2x+2) - 2) ] * (√(2x+2) + 2)/(√(2x+2) + 2)
= (x+1)(x-1)(√(2x+2) + 2)/(2x+2 - 4)
= (x+1)(x-1)(√(2x+2) + 2)/( 2(x-1))

= lim (x+1)(√(2x+2) + 2)/2 as x --> 1
= 2(√4+2)/2 = 4

Answer 2

=24