Let A be a subset of ℝ. Let c be a limit point of A. Consider the function f: A → ℝ Claim: If the function f has not have a limit at c, then there exists a sequence (xn), where xn≠c for all n, such that lim xn=c, but the sequence (f(xn)) does not converge. Since the function f does not have a limit at c, for all L in ℝ, there is some εL>0 such that for all δ>0 there is a point x in A satisfying 0<|x-c|<δ and yet |f(x)-L|≥εL. I know how to create a sequence that converges to 0; I am having trouble making the sequence (f(xn)) not converge. My initial idea was to pick points xn such that 0<|xn-c|<1/n and |f(xn)-n|≥εn for all n. However, this was not quite working out. Any bump in the right direction would be awesome!