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Left-hand and Right-hand Limits
A two-sided limit exists only when the left-hand and right-hand limits both exist and are equal.
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Left-hand and Right-hand Limits
What is it?
A two-sided limit exists only when the left-hand and right-hand limits both exist and are equal.
In simple words
- A two-sided limit exists only when the left-hand and right-hand limits both exist and are equal.
- Worked pattern: For sign(x) at 0, left limit is -1 and right limit is 1, so no limit exists.
- Final check: domain, sign, notation, units, and reasonableness.
Important terms
Left-hand and right-hand limits: A two-sided limit exists only when the left-hand and right-hand limits both exist and are equal.
What you should learn
- State the rule for left-hand and right-hand limits.
- Solve a left-hand and right-hand limits problem step by step.
- Check the result, restrictions, notation, and units.
Recommended for Class 12
Full school-level explanation
A two-sided limit exists only when the left-hand and right-hand limits both exist and are equal.
CONCEPT AND RULE
A two-sided limit exists only when the left-hand and right-hand limits both exist and are equal.
METHOD
For left-hand and right-hand limits, identify the given information and required result. Select the matching definition or formula, state its conditions, substitute carefully, simplify one line at a time, and check.
WORKED EXAMPLE
For sign(x) at 0, left limit is -1 and right limit is 1, so no limit exists.
COMMON MISTAKE
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
How to solve left-hand and right-hand limits
How it works
- Identify the mathematical objects, givens, restrictions, and required result.
- Choose and write the relevant definition, theorem, or formula.
- Substitute with brackets and carry out one justified step per line.
- Simplify, state the final answer clearly, and verify it in the original setting.
Remember
- Left-hand and right-hand limits
- rule
- worked example
- worked example
How the ideas connect
Worked example: For sign(x) at 0, left limit is -1 and right limit is 1, so no limit exists.
Deeper understanding
Go beyond memorising Left-hand and Right-hand Limits. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked example: For sign(x) at 0, left limit is -1 and right limit is 1, so no limit exists.
Common mistakes
Exam focus
Core Idea
Rule / Formula
Method
Worked Example
Exam focus
Quick revision
- Left-hand and right-hand limits
- rule
- worked example
- Left-hand and right-hand limits: A two-sided limit exists only when the left-hand and right-hand limits both exist and are equal.
- CONCEPT AND RULE: A two-sided limit exists only when the left-hand and right-hand limits both exist and are equal.
- METHOD: For left-hand and right-hand limits, identify the given information and required result. Select the
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
For long answers on Left-hand and Right-hand Limits: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- Left-hand and right-hand limits
- rule
- worked example
- Left-hand and right-hand limits: A two-sided limit exists only when the left-hand and right-hand limits both exist and are equal.
Important formulas
- Left-hand and right-hand limits
- rule
- worked example
Common exam traps
- Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
- A matching-looking formula is not valid until its conditions and domain have been checked.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
- Mistake
A matching-looking formula is not valid until its conditions and domain have been checked.
- Mistake
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
Left-hand and right-hand limits
rule
worked example
Left-hand and right-hand limits: A two-sided limit exists only when the left-hand and right-hand limits both exist and are equal.
A two-sided limit exists only when the left-hand and right-hand limits both exist and are equal.
- Definition
- A two-sided limit exists only when the left-hand and right-hand limits both exist and are equal.
- Critical warning
- Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all d
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Exam Practice
Define
Q1. Define Left-hand and Right-hand Limits.
Explain
Q2. Explain: CONCEPT AND RULE.
Example
Q3. Give one clear example of Left-hand and Right-hand Limits and say why it fits.
Mistake
Q4. State one common mistake students make about Left-hand and Right-hand Limits.
Exam note
Q5. Write a short exam note on Left-hand and Right-hand Limits: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.