Limits – Differential Calculus for GATE, ESE, AE/JE

Limits form the foundation of Differential Calculus and are extremely important for GATE, ESE, AE/JE exams. Understanding limits is essential before learning continuity and differentiation.

Basic Concept

The limit of a function describes the value that a function approaches as the input approaches a certain point.

lim (x → a) f(x) = L

This means f(x) approaches L as x approaches a.

Standard Limits (VERY IMPORTANT)

• lim (x → 0) (sin x / x) = 1
• lim (x → 0) ((1 − cos x) / x²) = 1/2
• lim (x → 0) (tan x / x) = 1
• lim (x → 0) ((eˣ − 1) / x) = 1
• lim (x → 0) (ln(1 + x) / x) = 1

Important Forms

Limits often appear in indeterminate forms:

• 0 / 0
• ∞ / ∞
• 0 × ∞
• ∞ − ∞
• 0⁰, ∞⁰, 1^∞

Methods to Solve Limits

1. Direct Substitution

If function is continuous, substitute directly.

2. Factorization

Used when 0/0 form appears.

3. Rationalization

Used when roots are present.

4. L’Hospital Rule

If limit is 0/0 or ∞/∞:

lim (f/g) = lim (f’/g’)

5. Standard Limits

Apply known results to simplify quickly.

Solved Examples

Example 1

lim (x → 0) (sin x / x)

Answer: 1

Example 2

lim (x → 2) (x² − 4) / (x − 2)

= (x − 2)(x + 2) / (x − 2)
= x + 2

Answer: 4

Example 3

lim (x → 0) ((eˣ − 1) / x)

Answer: 1

Example 4

lim (x → 0) (1 − cos x) / x²

Answer: 1/2

Shortcut Tricks

• Memorize all standard limits
• Convert into known forms
• Use expansion for higher level questions
• Apply L’Hospital only when necessary

Questions

Q1. lim (x → 0) (tan x / x)?

Answer: 1

Q2. lim (x → 0) (ln(1+x)/x)?

Answer: 1

Exam Tips

• Focus on standard limits first
• Identify indeterminate forms quickly
• Avoid unnecessary L’Hospital use
• Practice PYQs regularly

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