#lim_{x to 2} (x^2+2x-8)/(sqrt(x^2+5)-(x+1))#
#= lim_{x to 2} ((x+4)(x-2))/(sqrt(x^2+5)-(x+1))#
let #x = 2 + h, 0< abs(h) "<<" 0#
so the limit becomes
#= lim_{h to 0} ((6+h)(h))/(sqrt(9 + 4h + h^2)-(3+h))#
by Binomial expansion
#sqrt(9 + 4h + h^2)#
#= 3 (1 + 1/9 (4h+h^2) )^(1/2)#
#= 3 (1 + 1/2* 1/9 (4h+h^2) + O(h^2) )#, remembering that # 0< abs(h) "<<" 0#
#= 3 + 2/3 h + O(h^2) #
#implies lim_{h to 0} ((6+h)(h))/(3 + 2/3 h + O(h^2) -(3+h))#
#= lim_{h to 0} ((6+h)(h))/(-1/3 h + O(h^2) )#
#= lim_{h to 0} - (3(6+h))/(1 + O(h))#
#= -18#
