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.
Panhellenic Exams / Mathematics / Two-unit revision pilot
Chapter 1: Limits and continuity
Chapter 1: Limits and continuity: structured theory, worked examples, answered practice, and a mastery checklist for Panhellenic Exams.
CHAPTER PLAN
Learn, practise, and check your progress
Follow the steps in order or jump directly to the part you need.
—/5
completed
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.
Understanding the concept
A limit describes values near a point, not necessarily the value at that point. The form 0/0 is not an answer; it calls for a valid transformation, such as factorisation away from the point.
Method and conditions
The two finite one-sided limits must agree for the corresponding two-sided limit to exist. Continuity at a requires f(a) to be defined, the limit to exist, and that limit to equal f(a).
Checking and coverage boundaries
For Bolzano's theorem, check continuity on closed [a,b] and opposite signs at the endpoints. This proves at least one root in (a,b), not uniqueness. Uniqueness needs an extra argument, such as strict monotonicity.
Original pilot material — human educator review pending
Written specifically for YourFavTeacher with AI assistance, without copying past-paper prompts. Technical and numerical checks do not replace review by a human educator. It is not an official paper or endorsed by an examination body.
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
Worked example 3
If f(6)=5, can that fact alone establish lim(x→6)f(x)=5?
- 1A limit concerns nearby values, not only the value at the point.
- 2If f(x)=0 for x≠6 and f(6)=5, the limit is 0.
- 3This is a counterexample to the claim.
Not without additional information, such as continuity.
Chapter 1: Limits and continuity
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.
- Treating 0/0 as a number.
- Ignoring one of the one-sided limits.
- Claiming a unique root from Bolzano alone.
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.
Chapter 1: Limits and continuity
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
Panhellenic strategy: theory, methodology, timing, grading, and continuous correction.
What this chapter covers
The structure follows the official textbook layout and is used to organise study.
Limits and indeterminate forms
One-sided limits and continuity
Conditions of Bolzano's theorem
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.