#"The Reqd.Lim.="lim_(x to 1){sqrt(3-x)-2}/{sqrt(2-x)-1}#,
#=lim{sqrt(3-x)-2}/color(red){sqrt(2-x)-1}xx{sqrt(2-x)+1}/color(red){sqrt(2-x)+1}#,
#=lim{(sqrt(3-x)-2)(sqrt(2-x)+1)}/color(red){(2-x)-1}#,
#=lim{(sqrt(3-x)-2)(sqrt(2-x)+1)}/(1-x)#,
#=limcolor(blue)(sqrt(3-x)-2)xxcolor(blue)(sqrt(3-x)+2)/(sqrt(3-x)+2)*(sqrt(2-x)+1)/(1-x)#,
#lim{color(blue){(3-x)-2^2}*(sqrt(2-x)+1)}/{(sqrt(3-x)+2)(1-x)}#,
#lim{-(1+x)(sqrt(2-x)+1)}/{(sqrt(3-x)+2)(1-x)}#.
So, as long as #(1-x)# continues to occupy the Denominator,
the limit can not exist.
BONUS :
Had it been #lim_(x to 1){sqrt(3-x)-color(green)sqrt2}/{sqrt(2-x)-1}#, the Limit,
#=lim_(x to 1){((3-x)-2)(sqrt(2-x)+1)}/{(sqrt(3-x)+sqrt2)(1-x)}#,
#=lim{cancel((1-x))(sqrt(2-x)+1)}/{(sqrt(3-x)+sqrt2)cancel((1-x))}#,
#=lim_(x to 1)(sqrt(2-x)+1)/(sqrt(3-x)+sqrt2)#,
#=(sqrt(2-1)+1)/(sqrt(3-1)+sqrt2)#,
#=(1+1)/(sqrt2+sqrt2)#,
#=2/(2sqrt2)#,
#=1/sqrt2#,
#=sqrt2/2#.
