How do you evaluate the limit #(1-sqrtx)/(x-1)# as x approaches #1#?

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Answer 1 · 2016-09-07T08:52:21

#-1/2#

#lim_(x to 1) (1-sqrtx)/(x-1)#

let #x = 1 + delta#

#implies lim_(delta to 0) (1-sqrt(1 + delta))/(1 + delta -1)#

by Binomial Expansion

#= lim_(delta to 0) (1-(1 + 1/2 delta + O(delta^2)))/(delta)#

#= lim_(delta to 0) ( - 1/2 delta + O(delta^2))/(delta) = -1/2#

Answer 2 · 2016-09-07T09:08:03

#-1/2#.

The Reqd. Lim.#=lim_(xrarr1) (1-sqrtx)/(x-1)#

#=-lim_(xrarr1) (1-sqrtx)/(1-(sqrtx)^2)#

#=-lim_(xrarr1) cancel((1-sqrtx))/(cancel((1-sqrtx))(1+sqrtx))#

#=-lim_(xrarr1) 1/(1+sqrtx)#

#=-1/(1+sqrt1)#

#=-1/2#, as derived before by Respected Eddie!