学习目标
- I can evaluate limits using an appropriate transformation.
- I can check one-sided limits and continuity.
- I can apply the intermediate value theorem with its assumptions.
Panhellenic Exams / Mathematics / Curriculum
Κεφάλαιο 1: Όριο - Συνέχεια συνάρτησης
Κεφάλαιο 1: Όριο - Συνέχεια συνάρτησης: structured theory, worked examples, answered practice, and a mastery checklist for Panhellenic Exams.
章节计划
学习、练习并检查进度
按顺序完成步骤，或直接跳到你需要的部分。
—/5
已完成
预计主动学习时间
181 分钟
1. 理解
学习目标、核心概念与结构化理论。
打开 →2. 掌握方法
逐步说明的完整例题。
打开 →3. 练习
含提示与答案的分级练习。
打开 →4. 巩固
从基础到挑战的课后任务。
打开 →5. 检查
含计时、分值与解答的章节测试。
打开 →
进度只保存在当前浏览器中，不需要 AI credits。
核心概念与检查
- A limit describes what f(x) approaches, not necessarily f(x₀).
- A finite two-sided limit exists when both one-sided limits exist and agree.
- Continuity at x₀ requires f(x₀) to be defined and lim(x→x₀)f(x)=f(x₀).
单元
核心理论
练习前按清晰顺序掌握本章核心概念。
Limits and domain
Approach the point through the function's domain. A function may have a limit at a point where the function itself is undefined.
The indeterminate form 0/0
0/0 is not an answer. For rational expressions, factor and cancel for x≠x₀. For radicals, rationalisation may help. Then evaluate the equivalent expression's limit.
Continuity and the intermediate value theorem
For continuity, check both limit and value. A function continuous on [a,b] with opposite endpoint signs has at least one root in (a,b) by the intermediate value theorem, not necessarily a unique root.
Mathematics
例题讲解
逐步查看方法，并理解每一步为何成立。
A limit by factorisation
Evaluate lim(x→2) (x²−4)/(x−2).
- 1Direct substitution gives 0/0, so transform the expression.
- 2x²−4=(x−2)(x+2).
- 3For x≠2, cancel to get x+2. As x→2, x+2 tends to 4.
4
Continuity at a point
For x≠2, f(x)=(x²−4)/(x−2). How should f(2) be defined for continuity?
- 1The limit at 2 is 4, by factorisation.
- 2Continuity requires the value to equal the limit, so define f(2)=4.
f(2)=4
Κεφάλαιο 1: Όριο - Συνέχεια συνάρτησης
含答案练习
八道分层练习，从基础熟练到考试应用。请先独立完成，再查看提示或答案。
基础熟练5 分钟
Find lim(x→2) (3x + 1).
提示
A polynomial is continuous: substitute for x.
答案指南
3·2 + 1 = 7
基础熟练6 分钟
Find lim(x→2) (x² − 4)/(x − 2).
提示
Factor the difference of squares and cancel only away from the limit point.
答案指南
For x ≠ 2, the quotient equals x + 2. The limit is 4.
应用7 分钟
Let f(x)=(x² − 4)/(x − 2) for x≠2, and f(2)=k. Which k makes f continuous?
提示
The function value must equal the limit.
答案指南
k = 4
应用8 分钟
For x<0, f(x)=3; for x≥0, f(x)=4. Does lim(x→0) f(x) exist?
提示
Calculate the two one-sided limits.
答案指南
No: left-hand 3, right-hand 4.
推理9 分钟
Find the one-sided limits of |x|/x at 0.
提示
For x<0, |x|=−x; for x>0, |x|=x.
答案指南
Left-hand −1, right-hand 1. The two-sided limit does not exist.
推理10 分钟
Find lim(x→4) (√x − 2)/(x − 4).
提示
Rationalise to obtain 1/(√x+2) for x≠4.
答案指南
1/4
考试题型11 分钟
Let f(x)=x+3 for x<2 and f(x)=2x+k for x≥2. Find k for continuity at 2.
提示
Equate the left limit and the second expression's value.
答案指南
2+3=2·2+k ⇒ k=1
考试题型12 分钟
Find lim(x→+∞) (3x²+2)/(2x²+1).
提示
Divide numerator and denominator by x².
答案指南
3/2
常见错误
- Giving 0/0 as a final answer.
- Cancelling without retaining x≠x₀.
- Confusing existence of a limit with continuity.
掌握度检查
- I can evaluate limits using an appropriate transformation.
- I can check one-sided limits and continuity.
- I can apply the intermediate value theorem with its assumptions.
Κεφάλαιο 1: Όριο - Συνέχεια συνάρτησης
章节作业
六道不同作业，从基础到挑战，包含预计用时、提示和答案指南。
基础8 分钟
Find lim(x→3) (4x + 1).
提示
A polynomial is continuous: substitute for x.
答案指南
4·3 + 1 = 13
基础8 分钟
Find lim(x→3) (x² − 9)/(x − 3).
提示
Factor the difference of squares and cancel only away from the limit point.
答案指南
For x ≠ 3, the quotient equals x + 3. The limit is 6.
进阶8 分钟
Let f(x)=(x² − 9)/(x − 3) for x≠3, and f(3)=k. Which k makes f continuous?
提示
The function value must equal the limit.
答案指南
k = 6
进阶8 分钟
For x<0, f(x)=4; for x≥0, f(x)=5. Does lim(x→0) f(x) exist?
提示
Calculate the two one-sided limits.
答案指南
No: left-hand 4, right-hand 5.
挑战8 分钟
Find the one-sided limits of |x|/x at 0.
提示
For x<0, |x|=−x; for x>0, |x|=x.
答案指南
Left-hand −1, right-hand 1. The two-sided limit does not exist.
挑战8 分钟
Find lim(x→9) (√x − 3)/(x − 9).
提示
Rationalise to obtain 1/(√x+3) for x≠9.
答案指南
1/6
50 分钟 / 40 分
完整章节测试
带计时器、完整分值和参考答案的章节自测。
Show your working and justify each theorem's assumptions. Check the answers after finishing.
1. Find lim(x→4) (5x + 1).
4 分答案指南
5·4 + 1 = 21
2. Find lim(x→4) (x² − 16)/(x − 4).
4 分答案指南
For x ≠ 4, the quotient equals x + 4. The limit is 8.
3. Let f(x)=(x² − 16)/(x − 4) for x≠4, and f(4)=k. Which k makes f continuous?
4 分答案指南
k = 8
4. For x<0, f(x)=5; for x≥0, f(x)=6. Does lim(x→0) f(x) exist?
4 分答案指南
No: left-hand 5, right-hand 6.
5. Find the one-sided limits of |x|/x at 0.
4 分答案指南
Left-hand −1, right-hand 1. The two-sided limit does not exist.
6. Find lim(x→16) (√x − 4)/(x − 16).
4 分答案指南
1/8
7. Let f(x)=x+5 for x<4 and f(x)=2x+k for x≥4. Find k for continuity at 4.
4 分答案指南
4+5=2·4+k ⇒ k=1
8. Find lim(x→+∞) (5x²+4)/(2x²+1).
4 分答案指南
5/2
9. Find lim(x→0) 5x/(√(1+x)+1).
4 分答案指南
0/2 = 0
10. Show that g(x)=x³+x−5 has at least one root in (0, 5).
4 分答案指南
g is continuous on [0, 5], g(0)=−5<0 and g(5)=125>0. The intermediate value theorem gives a root in (0, 5).
单元
官方来源与核验
课程参考来源；教学顺序请始终与学校和老师确认。
Mathematics
Panhellenic Exams: 数学与物理的结构化路线，包含章节、练习和下一步。
本章覆盖内容
结构跟随教材，用来帮助组织学习。
Πραγματικοί αριθμοί
Συναρτήσεις
Μονοτονία
Ακρότατα
Συμμετρίες
Αντίστροφη συνάρτηση
Όριο συνάρτησης στο `x0`
Ιδιότητες ορίων
Μη πεπερασμένο όριο
Όρια στο άπειρο
Συνέχεια συνάρτησης
Θεωρήματα συνεχών συναρτήσεων
重点关注
通常容易出错或需要更多复习的部分。
I can evaluate limits using an appropriate transformation.
I can check one-sided limits and continuity.
I can apply the intermediate value theorem with its assumptions.
来源、日常资料与学习资源
从教材、daily material、PDFs、videos 和 worked examples 开始。
Start from the official textbook or specification referenced on the subject page.
Use the notes and examples as support, not as a replacement for the official syllabus.
Check the current syllabus version before exam preparation.
按小主题练习
进入 full tests 前先做 targeted practice，让覆盖情况更清楚。
Find lim(x→2) (3x + 1).
Find lim(x→2) (x² − 4)/(x − 2).
Let f(x)=(x² − 4)/(x − 2) for x≠2, and f(2)=k. Which k makes f continuous?
For x<0, f(x)=3; for x≥0, f(x)=4. Does lim(x→0) f(x) exist?
Find the one-sided limits of |x|/x at 0.
Find lim(x→4) (√x − 2)/(x − 4).
Let f(x)=x+3 for x<2 and f(x)=2x+k for x≥2. Find k for continuity at 2.
Find lim(x→+∞) (3x²+2)/(2x²+1).
Mocks 与进度检查
如何衡量本章进度，以及何时加入综合 mock。
Start with untimed practice by subtopic.
Move to a short timed checkpoint only after completing the mastery checklist.
Record each error with the correct method and revisit it after 48 hours.
下一步
完成本章后该做什么，以及它如何连接到下一单元。
Complete the practice without support.
Explain the core method aloud in under two minutes.
Continue to the next chapter or request targeted tutor support.
说明：每学年的正式考试范围请始终以学校、老师和当前官方来源为准。