{"schema_version":"ocme-site-results-v1","locale":"en","page_type":"derivations","facts":{"questions":36000,"reviewed":36000,"display_candidates":0,"knowledge":302,"guides":299,"paths":59,"batches":360,"families":1441,"atlas":302,"core":80,"evidence":9,"formal_evidence":5,"formal_subjects":4,"lean_files":6,"lean_toolchain":"leanprover/lean4:v4.30.0","question_state":"candidate","english_state":"ai_candidate","question_Lean":"NOT_RUN","canonical_truth_promotion":false},"filters":{"q":"mko-function-limit"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-function-limit-01","title":{"zh-Hant":"函數極限：可去因子商、仿射 δ 界與不等的單側極限","en":"Function limits: a removable-factor quotient, affine δ bounds and unequal one-sided limits"},"summary":{"zh-Hant":"在去心實數論域上：(1) 以 ε-δ 證明 f(x)=(x²−9)/(x−3) 滿足 lim_(x→3) f(x)=6，而 f(3) 仍無定義；(2) 對仿射 g(x)=4x−1 在 a=2、ε=1/2，證明 δ=1/8 合法（等號 |m|δ=ε 允許），δ=1/7 不合法；(3) 對分支函數 h(x)=x+1（x<1）、h(x)=2x+3（x>1），求左右極限 2 與 5，並證明兩側極限不存在。","en":"On punctured real domains: (1) prove by ε-δ that f(x)=(x²−9)/(x−3) has lim_(x→3) f(x)=6 while f(3) stays undefined; (2) for the affine g(x)=4x−1 at a=2 with ε=1/2, prove that δ=1/8 is valid (the equality |m|δ=ε is allowed) and δ=1/7 is not; (3) for the branch function h(x)=x+1 (x<1), h(x)=2x+3 (x>1), find the one-sided limits 2 and 5 and prove that the two-sided limit does not exist."},"mko_ids":["mko-function-limit"]}],"title":"Concept derivation guides"}