I tutor high school students in Calc and the other day I came across this problem. Limit of (1-Sqrt(x-2))/(x-3) as x->3 I tried coaching the student on how to simplify the expression and in the end I just showed him this substitution. Let u=Sqrt(x-2) Then (1-Sqrt(x-2))/(x-3) = (1-u)/(u^2-1) And the Limit becomes Limit of (1-u)/(u^2-1)=-1/(u+1) as u->1 which is -1/2 He looked at me like I had just done some black magic. I explained substitution to him and why it worked, showed him a couple of other simple examples, and confirmed the answer numerically (like they do in basic calc books when the limit concept is first presented). I still don't think he is 100% convinced because they had not covered this in his class yet which leads to my question. Can the original problem be solved without substitution?