How do you evaluate the limit #(x^2+2x-8)/(sqrt(x^2+5)-(x+1))# as x approaches 2?

1 Answer
Aug 5, 2016

-18

Explanation:

#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#