JEE/Mathematics/Limits and Continuity

Calculus · Must Do

Limits and Continuity — JEE Main & Advanced Notes

Limits are local behaviour; continuity means value and limiting behaviour agree.

standard limitscontinuityLHL/RHLindeterminate forms
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1. Introduction & Exam Weightage

Limits are local behaviour; continuity means value and limiting behaviour agree.

Material signal: Mapped from limits and continuity-differentiability revision assignments.

Priority: Must Do. Treat this as a moderate chapter inside the JEE Mathematics ladder.

2. Core Concepts & Definitions

  • Limit depends on nearby values, not necessarily the function value at the point.
  • Continuity requires LHL = RHL = f(a).
  • Indeterminate forms demand simplification or expansion.

3. Key Formulas with Derivation Hints

  • lim_{x o0} rac{sin x}{x}=1
    Hint: do not memorise this in isolation; connect it to the definition or diagram that produces it.
  • lim_{x o0} rac{e^x-1}{x}=1
    Hint: do not memorise this in isolation; connect it to the definition or diagram that produces it.
  • f(a)=lim_{x o a}f(x) for continuity
    Hint: do not memorise this in isolation; connect it to the definition or diagram that produces it.
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4. Solved Examples — Original Practice Models

Limits and Continuity: main-style warm-up

Problem: A direct one-step Limits and Continuity question is solved by identifying the governing definition before substituting values.

Method: write the target quantity, list the known values, use the standard relation, and check if the answer is dimensionally or logically possible. This prevents option-led guessing.

Limits and Continuity: advanced-style trap

Problem: A multi-condition Limits and Continuity problem often has an extra restriction hidden in domain, sign, interval or geometry.

Method: solve algebraically first, then filter using the domain/interval/geometry condition. In JEE Advanced, most wrong options come from skipping this final filtering step.

5. Common Mistakes & Traps

  • Substituting blindly into 0/0.
  • Checking only one side for piecewise functions.
  • Using degree comparison after cancellation is needed.

6. JEE Main Specific Strategy

For JEE Main, aim for fast recognition and clean substitution. Finish the first pass of Limits and Continuity questions in 60–90 seconds each. Prioritise standard formulas, short sign/domain checks and option elimination only after the setup is correct.

7. JEE Advanced Specific Strategy

For JEE Advanced, expect combined conditions, hidden domains, multi-correct traps and integer-style answers. Build the solution from definitions, not memorised tricks. When a parameter appears, solve the general case and then filter using restrictions.

8. Quick Revision Summary

  • Check LHL/RHL.
  • Use standard limits.
  • Simplify before substituting.
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