MathematicsChapter 2: Limit, Continuity and Derivative
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.
Learning objectives
- 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.
Definition
A two-sided limit exists only when the left-hand and right-hand limits both exist and are equal.
Where / symbols
- 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 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
- 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.
Worked example
- Worked example:
- For sign(x) at 0, left limit is -1 and right limit is 1, so no limit exists.
In simple words
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
Key points
- Left-hand and right-hand limits
- rule
- worked example
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Left-Hand and Right-Hand Limits
Sticky Notes · Formula and Worked Revision
Core Idea
- A two-sided limit exists only when the left-hand and right-hand limits both exist and are equal.
Rule / Formula
- A two-sided limit exists only when the left-hand and right-hand limits both exist and are equal.
Method
- Identify givens and restrictions.
- Write the rule.
- Substitute and simplify.
- Check the result.
Worked Example
- For sign(x) at 0, left limit is -1 and right limit is 1, so no limit exists.
Common Error
- Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
FBISE Exam Tip
- Show the formula, substitutions, algebra, restrictions, and final answer; method marks need visible reasoning.
60-second revision
- 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.
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