Introduction to Limits for JEE Main

The chapter Limits is the first fundamental topic of Calculus in Class 11 Maths, forming the foundation for differentiation and integration in JEE Main.

A limit describes the value that a function approaches as the input approaches a certain point. Understanding limits, algebra of limits, standard limits, and continuity is essential for solving numerical, conceptual, and application-based problems efficiently.

Direct Links to Download JEE Main - Chapter Wise Previous Year Papers Maths (PDF)

► Click “Download Here” next to your subject to access the free PDF.

Definition of Limit

1. Basic Definition:

• For a function f(x), the limit as x approaches a is L, written as:
  lim(x → a) f(x) = L

2. Intuitive Meaning:

• As x gets closer to a, the value of f(x) gets closer to L.

3. Left-Hand Limit (LHL) and Right-Hand Limit (RHL):

• LHL: lim(x → a⁻) f(x)

• RHL: lim(x → a⁺) f(x)

• Existence Condition: LHL = RHL = Limit

Algebra of Limits

1. Sum Rule: lim(f(x) + g(x)) = lim f(x) + lim g(x)

2. Difference Rule: lim(f(x) – g(x)) = lim f(x) – lim g(x)

3. Product Rule: lim(f(x) × g(x)) = (lim f(x)) × (lim g(x))

4. Quotient Rule: lim(f(x)/g(x)) = (lim f(x)) / (lim g(x)), g(x) ≠ 0

5. Constant Multiple: lim(k × f(x)) = k × lim f(x)

Standard Limits

1. lim(x → 0) (sin x)/x = 1

2. lim(x → 0) (1 – cos x)/x = 0

3. lim(x → 0) (1 + x)^(1/x) = e

4. lim(x → 0) (tan x)/x = 1

• These standard limits are frequently used in JEE Main problems

Techniques of Evaluating Limits

1. Factorization Method:

• Factor numerator/denominator to simplify and cancel common terms

2. Rationalization Method:

• Multiply numerator/denominator by conjugate for roots

3. Substitution Method:

• Direct substitution if f(a) is defined

4. L’Hospital’s Rule (for 0/0 or ∞/∞ forms):

• lim(x → a) f(x)/g(x) = lim(x → a) f'(x)/g'(x)

Limits Involving Infinity

1. Horizontal Asymptotes:

• lim(x → ∞) f(x) or lim(x → –∞) f(x)

• If degree of numerator < denominator → 0

• If degrees equal → ratio of leading coefficients

2. Vertical Asymptotes:

• Points where denominator = 0

Continuity of a Function

1. Definition:

• A function f(x) is continuous at x = a if:

  • f(a) is defined

  • lim(x → a) f(x) exists

  • lim(x → a) f(x) = f(a)

2. Types of Discontinuity:

• Removable, Jump, Infinite

3. Application:

• Continuity ensures smooth graphs, required for derivatives and integration

Applications of Limits in JEE Main

• Basis for differentiation and integration problems

• Solve maxima-minima and rate of change problems

• Analyze function behavior near a point

• Frequently tested in MCQs, integer-type, and numerical problems

Example: Evaluate lim(x → 0) (sin 3x)/x → 3 (using standard limit sin x/x → 1)

Important JEE Main Preparation Tips for Limits

• Memorize standard limits and algebraic properties

• Practice evaluating limits using factorization, rationalization, and substitution

• Solve L’Hospital Rule problems for indeterminate forms

• Solve previous year JEE Main MCQs and numerical problems

• Visualize graphical approach for continuity and limit problems

Why Study Limits from Studentbro.in?

Studentbro.in provides:

• Step-by-step explanations of definition, standard limits, and L’Hospital Rule

• Solved examples for algebraic, trigonometric, and exponential functions

• Charts and tables for quick revision of standard limits

• MCQs and PYQs aligned with JEE Main syllabus

Conclusion

The Limits chapter is the foundation of Calculus in Class 11 Maths for JEE Main. Mastery of standard limits, algebra of limits, continuity, and L’Hospital Rule helps students solve conceptual and numerical problems efficiently.

Studentbro.in provides structured, easy-to-understand, and exam-focused content to master Limits effectively and boost JEE Main scores.