I'm sorry.....what are you saying?
I understand what a limit is....but i'm not certain how to state this as a formal proof as it relates to the question.
I know if we graph the function it is very easy to see that it approaches 1 as x->1.....but how do you prove it?
I'm also not sure how the example you have given applies to this question?
let us try the proof by contradiction way.
let us suppose that no limit exists for sqrt(x) ie. the expression doesnot converge to any value ie. it doesnot have any bounds. this is false. so perhaps something should come out. you could also go the way you suggested yourself. but using convergence and series willl be a better way because the concept of limit has come from it.