lim as x goes to 1 (x^(2)-2x+4)=3
Using what "basis"? If you know about "continuous functions" and in particular that all polynomials are continous for all values of x then it is sufficient to say, as emakarov does, that the limit is $\displaystyle (1)^2-2(1)+ 4= 1- 2+ 4= 3$.
If, on the other hand, you have only the definition of limit to work with, you need to look at $\displaystyle |f(x)- L|= |x^2- 2x+ 4- 3|= |x^2- 2x+ 1|= |x-1|^2< \epsilon$, then what can you say if you choose $\displaystyle \delta= \sqrt{\epsilon}$ so that $\displaystyle |x- 1|< \delta$ becomes $\displaystyle |x- 1|< \sqrt{\epsilon}$?