Learning objectives
- 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.
Greek Lyceum Grade 3 / Mathematics / Curriculum
Όρια και συνέχεια
Όρια και συνέχεια: structured theory, worked examples, answered practice, and a mastery checklist for Greek Lyceum Grade 3.
CHAPTER PLAN
Learn, practise, and check your progress
Follow the steps in order or jump directly to the part you need.
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Estimated active study time
181 minutes
1. Understand
Objectives, key ideas, and structured theory.
Open →2. Follow the method
Worked examples that explain every step.
Open →3. Practise
Graded tasks with hints and answer guidance.
Open →4. Consolidate
Assignments from core work to challenge.
Open →5. Check
A timed, marked test with solution guidance.
Open →
Progress is stored only in this browser and does not require AI credits.
Key ideas and checks
- 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).
Unit
Core theory
The essential chapter ideas in a clear sequence before practice.
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
Worked examples
Follow the method step by step and check why every step is valid.
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
Όρια και συνέχεια
Practice with answers
Eight graded tasks from core fluency to exam-style application. Work independently before opening a hint or answer.
Core fluency5 minutes
Find lim(x→2) (3x + 1).
Hint
A polynomial is continuous: substitute for x.
Answer guide
3·2 + 1 = 7
Core fluency6 minutes
Find lim(x→2) (x² − 4)/(x − 2).
Hint
Factor the difference of squares and cancel only away from the limit point.
Answer guide
For x ≠ 2, the quotient equals x + 2. The limit is 4.
Application7 minutes
Let f(x)=(x² − 4)/(x − 2) for x≠2, and f(2)=k. Which k makes f continuous?
Hint
The function value must equal the limit.
Answer guide
k = 4
Application8 minutes
For x<0, f(x)=3; for x≥0, f(x)=4. Does lim(x→0) f(x) exist?
Hint
Calculate the two one-sided limits.
Answer guide
No: left-hand 3, right-hand 4.
Reasoning9 minutes
Find the one-sided limits of |x|/x at 0.
Hint
For x<0, |x|=−x; for x>0, |x|=x.
Answer guide
Left-hand −1, right-hand 1. The two-sided limit does not exist.
Reasoning10 minutes
Find lim(x→4) (√x − 2)/(x − 4).
Hint
Rationalise to obtain 1/(√x+2) for x≠4.
Answer guide
1/4
Exam style11 minutes
Let f(x)=x+3 for x<2 and f(x)=2x+k for x≥2. Find k for continuity at 2.
Hint
Equate the left limit and the second expression's value.
Answer guide
2+3=2·2+k ⇒ k=1
Exam style12 minutes
Find lim(x→+∞) (3x²+2)/(2x²+1).
Hint
Divide numerator and denominator by x².
Answer guide
3/2
Common mistakes
- Giving 0/0 as a final answer.
- Cancelling without retaining x≠x₀.
- Confusing existence of a limit with continuity.
Mastery check
- 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.
Όρια και συνέχεια
Chapter homework
Six distinct assignments from core fluency to challenge, each with an estimated time, hint, and answer guide.
Core8 minutes
Find lim(x→3) (4x + 1).
Hint
A polynomial is continuous: substitute for x.
Answer guide
4·3 + 1 = 13
Core8 minutes
Find lim(x→3) (x² − 9)/(x − 3).
Hint
Factor the difference of squares and cancel only away from the limit point.
Answer guide
For x ≠ 3, the quotient equals x + 3. The limit is 6.
Stretch8 minutes
Let f(x)=(x² − 9)/(x − 3) for x≠3, and f(3)=k. Which k makes f continuous?
Hint
The function value must equal the limit.
Answer guide
k = 6
Stretch8 minutes
For x<0, f(x)=4; for x≥0, f(x)=5. Does lim(x→0) f(x) exist?
Hint
Calculate the two one-sided limits.
Answer guide
No: left-hand 4, right-hand 5.
Challenge8 minutes
Find the one-sided limits of |x|/x at 0.
Hint
For x<0, |x|=−x; for x>0, |x|=x.
Answer guide
Left-hand −1, right-hand 1. The two-sided limit does not exist.
Challenge8 minutes
Find lim(x→9) (√x − 3)/(x − 9).
Hint
Rationalise to obtain 1/(√x+3) for x≠9.
Answer guide
1/6
50 minutes / 40 marks
Full chapter test
A timed, full-mark self-assessment with model-answer guidance.
Show your working and justify each theorem's assumptions. Check the answers after finishing.
1. Find lim(x→4) (5x + 1).
4 marksAnswer guide
5·4 + 1 = 21
2. Find lim(x→4) (x² − 16)/(x − 4).
4 marksAnswer guide
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 marksAnswer guide
k = 8
4. For x<0, f(x)=5; for x≥0, f(x)=6. Does lim(x→0) f(x) exist?
4 marksAnswer guide
No: left-hand 5, right-hand 6.
5. Find the one-sided limits of |x|/x at 0.
4 marksAnswer guide
Left-hand −1, right-hand 1. The two-sided limit does not exist.
6. Find lim(x→16) (√x − 4)/(x − 16).
4 marksAnswer guide
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 marksAnswer guide
4+5=2·4+k ⇒ k=1
8. Find lim(x→+∞) (5x²+4)/(2x²+1).
4 marksAnswer guide
5/2
9. Find lim(x→0) 5x/(√(1+x)+1).
4 marksAnswer guide
0/2 = 0
10. Show that g(x)=x³+x−5 has at least one root in (0, 5).
4 marksAnswer guide
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).
Unit
Official sources and verification
Curriculum reference sources. Always confirm the teaching sequence with the school and tutor.
Mathematics
A full programme: official syllabus, theory, methodology, topic exercises, mocks, and systematic error correction.
What this chapter covers
The structure follows the official textbook layout and is used to organise study.
Όριο σε σημείο
Πλευρικά όρια
Πράξεις ορίων
Άπειρα όρια
Όρια στο άπειρο
Συνέχεια
Θεώρημα Bolzano
Θεώρημα ενδιάμεσων τιμών
Θεώρημα μέγιστης και ελάχιστης τιμής
Where to focus
The areas that usually create mistakes or need extra revision.
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.
Sources, daily material, and resources
Where to start: textbook, daily material, PDFs, videos, and worked examples.
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.
Practice by subtopic
Targeted practice before full tests so coverage is clear.
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 and progress checks
How to measure progress in this chapter and when it enters a cumulative 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.
Next step
What to do after finishing the chapter and how it connects to the next unit.
Complete the practice without support.
Explain the core method aloud in under two minutes.
Continue to the next chapter or request targeted tutor support.
Note: for the official examinable syllabus of each school year, always confirm with the school, tutor, and current Ministry/IEP announcements.