Finding limit of #(sqrt(3-x)-2)/(sqrt(2-x)-1)# as x approaches 1?

1 Answer
Feb 21, 2018

Kindly refer to the Discussion in the Explanation.

Explanation:

#"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#.