Limits — Piecewise & LHL/RHL explained through a friendly whiteboard conversation, examples, visuals, and practice for school students.
Step 1
Notes
Step 2
Formulae
Step 3
Examples
Step 4
Practice
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A limit describes the value a function approaches near a point.
Piecewise functions need special care because the rule may change at the point.
Limits are the foundation for continuity, derivatives, and calculus.
A lefand limit checks values approaching from smaller alues.
A righand limit checks values approaching from larger alues.
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For a piecewise function, use the rule that matches the side of approach.
The value of the function at the point may be different from the limit.
Meaning: A limit is the value a function gets close to.
Definition: .
The limit focuses on nearby values, not only the value at .
Meaning: Approach from the left side.
Definition: LHL means x approaches a from values less than a.
Use the piece valid for .
Meaning: Approach from the right side.
Definition: RHL means x approaches a from values greater than a.
Use the piece valid for .
Meaning: A function is continuous when there is no break at the point.
Definition: Continuity at a needs the limit to exist and equal f(a).
Check LHL, RHL, and the function value.
Easy
Question: If LH and RH, what is the limit?
Medium
Question: If LH and RH at , does the limit exist?
Exam-level
Question: A function has limit 3 at but . Is it continuous at ?
Using the wrong piece of a piecewise function
Why it happens: The side of approach is ignored.
Correct approach: Use the rule for for LHL and for RHL.
Thinking the function value is always the limit
Why it happens: The point at gets all attention.
Correct approach: Limits depend on nearby values.
Saying a limit exists when LHL and RHL differ
Why it happens: Only one side was checked.
Correct approach: Both onided limits must match.
Can a limit exist if f(a) is different?
Yes, but the function is not continuous at that point.
Why do piecewise functions need LHL and RHL?
Because different rules may apply on different sides.
What if LHL and RHL are not equal?
The twided limit does not exist.
Practise the same concept from this article. Your student profile may guide entitlement and follop recommendations, but it does not replace this concept.