# Concept derivation guides

299 concept teaching guides, separate from the 36,000 individual question derivations.

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Concept guide



## [Tends-to relation: x→a is not x=a, and does not guarantee that function values converge](/en/derivations/derivation-mko-tends-to-relation-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-tends-to-relation)

In (ℝ, d(x,y)=|x−y|): (1) prove that the sequence x_n=1+1/n tends to 1 while every term satisfies x_n≠1; (2) for g(x)=1 (x>1) and g(x)=0 (x≤1), prove that g(x_n)→1 along x_n, but lim_{x→1} g(x) does not exist. This shows that x→1 by itself neither requires x to equal 1 nor decides whether function values converge.

Tends-to relation

derivation-mko-tends-to-relation-01





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