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A. Let f: R--->R and let c be element in R. Show that the lim from x to c of f(x)=L if and only if lim from x to 0 of f(x+c)=L
(if and only if: go both ways)

b. Use either the epsilon-delta definition (which states: Let A be a subset of the reals and let c be a cluster point of A. For a function f: A--->R, a real number L is said to be a limit of f at c if, given any epsilon>0 there exists a delta>0 such that if x is an element of A and 0<abs(x-c)<delta then abs(f(x)-L)< epsilon) or the sequential criterion for limits(which states: Let f: A-->Reals(R) and let c be a cluster point of A, then the following are equivelent i. lim as x goes to c of f=L and ii. for every sequence Xsubn in A that converges to c such that Xsubn does not equal c for all n element N, the sequence (f(Xsubn)) converges to L)
to establish the following limits

i. lim as x goes to 2 of (1/(1-x))= -1
ii. lim as x goes to 1 of (x/(1+x))= 1/2
iii. lim as x goes to 0 of ((x^2)/abs(x))=0
iv. lim as x goes to 1 of ((x^2-x+1)/(x+1))=1/2

Solution Summary

This provides an example of completing a proof regarding limits and examples of using a definition of limits to establish limits of functions.