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NDA Limits & Continuity

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This page covers NDA Limits & Continuity with complete concept notes, 42 graded practice MCQs, key points and exam-specific tips. Free to study.

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Concept Notes

Limits & Continuity— Rules & Concept

1.Core Concept

Read this first — the foundation of the topic
Think of it as asking

'What value does f(x) get closer to as x gets closer to a specific number?' The limit may exist even when the function is not defined at that point. The formal notation is: lim(x→a) f(x) = L, meaning as x approaches 'a', f(x) approaches 'L'

KEY LIMIT RULES

1. lim(x→a) k = k (constant rule) 2. lim(x→a) x = a (identity rule) 3. lim(x→a) [f(x) + g(x)] = lim(x→a) f(x) + lim(x→a) g(x) 4. lim(x→a) [f(x) × g(x)] = lim(x→a) f(x) × lim(x→a) g(x) 5. lim(x→a) [f(x)/g(x)] = lim(x→a) f(x) / lim(x→a) g(x), provided lim(x→a) g(x) ≠ 0 IMPORTANT STANDARD LIMITS: • lim(x→0) (sin x)/x = 1 • lim(x→0) (tan x)/x = 1 • lim(x→0) (1-cos x)/x² = 1/2 • lim(x→∞) (1 + 1/x)^x = e CONTINUITY means a function has no breaks, jumps, or holes at a point. A function f(x) is continuous at x = a if: 1. f(a) is defined 2. lim(x→a) f(x) exists 3. lim(x→a) f(x) = f(a)

2.Exam Patterns

What examiners ask — read before attempting PYQs

NDA typically asks direct limit evaluation (40%), continuity problems (30%), and application-based questions (30%). Most questions are 2-3 marks. SHORTCUT FOR 0/0 FORMS: When direct substitution gives 0/0, try factoring, rationalization, or L'Hospital's rule. For polynomial ratios, factor common terms first.

3.Worked Example

Solve this step-by-step before moving on
1
Step 1

Direct substitution gives (4-4)/(2-2) = 0/0 (indeterminate)

2
Step 2

Factor numerator: x²-4 = (x-2)(x+2)

3
Step 3

Simplify: (x²-4)/(x-2) = (x-2)(x+2)/(x-2) = x+2

4
Step 4

Now substitute: lim(x→2) (x+2) = 2+2 = 4 Answer: 4 WORKED EXAMPLE 2: Find lim(x→0) (sin 3x)/(2x)

1
Step 1

Rewrite as lim(x→0) (sin 3x)/(2x) = (3/2) × lim(x→0) (sin 3x)/(3x)

2
Step 2

Let u = 3x, so as x→0, u→0

3
Step 3

This becomes (3/2) × lim(u→0) (sin u)/u = (3/2) × 1 = 3/2 Answer: 3/2

4.Shortcuts

Use these to save 30–60 seconds per question

For lim(x→0) (sin ax)/(bx), the answer is always a/b. This saves 2-3 minutes in exams! THE

5.Exam Traps

Common mistakes students make — avoid these

Students often conclude that if direct substitution gives 0/0, the limit doesn't exist. This is WRONG! The 0/0 form means you need to simplify further using algebraic manipulation or standard limits.

The limit may still exist and have a finite value. Another frequent error is confusing the existence of a limit with the function being defined at that point. A limit can exist even when the function is undefined at that specific point.

Key Points to Remember

  • A limit is the value a function approaches as input approaches a specific point
  • Standard limit: lim(x→0) (sin x)/x = 1 - memorize this formula
  • Function is continuous at x=a if f(a) exists, limit exists, and both are equal
  • Quick formula: lim(x→0) (sin ax)/(bx) = a/b saves time in exams
  • 0/0 form means simplify further, not that limit doesn't exist
  • Factor polynomials when direct substitution gives indeterminate forms
  • Standard limit: lim(x→0) (1-cos x)/x² = 1/2 - high-frequency formula
  • Left hand limit must equal right hand limit for limit to exist
  • Rationalization technique works well for square root expressions in limits
  • L'Hospital's rule: if f'(a) and g'(a) exist, lim(x→a) f(x)/g(x) = f'(a)/g'(a) for 0/0 forms

Exam-Specific Tips

  • lim(x→0) (sin x)/x = 1 is the most frequently tested standard limit
  • lim(x→0) (tan x)/x = 1 appears in 60% of NDA limit questions
  • lim(x→0) (1-cos x)/x² = 1/2 is tested in trigonometric limit problems
  • e = lim(x→∞) (1 + 1/x)^x ≈ 2.718 is Euler's number definition
  • Polynomial functions are continuous everywhere on their domain
  • Rational functions are discontinuous only where denominator equals zero
  • lim(x→∞) (1/x) = 0 is the standard infinity limit
  • If lim(x→a⁺) f(x) ≠ lim(x→a⁻) f(x), then limit at x=a does not exist
Practice MCQs

Limits & Continuity — Practice Questions

42graded MCQs · easy to hard · full solution & trap analysis · showing 20 of 42

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Practice 1easy

Evaluate lim(x→0) [sin(3x)]/x.

Practice 2easy

Determine the value of lim(x→2) [x³ − 8]/(x − 2).

Practice 3easy

Evaluate lim(x→0) [sin(3x) / x].

Practice 4easy

Let f(x) = |x − 3|. At which point is f(x) discontinuous?

Practice 5easy

Determine whether f(x) = (x² - 4)/(x - 2) is continuous at x = 2.

Practice 6easy

Find lim(x→∞) [(2x² + 3x + 1) / (5x² - 2x + 4)].

Practice 7easy

Let f(x) = |x - 2|. At which point is f(x) discontinuous?

Practice 8easy

Evaluate lim(x→∞) [5x² + 3x + 2]/(2x² − x + 1).

Practice 9easy

Determine the value of lim(x→2) [(x³ − 8) / (x − 2)].

Practice 10easy

Let f(x) = |x − 3|. Determine whether f is continuous at x = 3.

Practice 11easy

Evaluate lim(x→∞) [(2x² + 5x + 1) / (3x² − 4x + 2)].

Practice 12easy

Let f(x) = (x² + 3x + 2)/(x + 1) for x ≠ -1. Find lim(x→-1) f(x).

Practice 13easy

Determine the value of lim(x→0) [sin(3x) / (5x)].

Practice 14easy

Let f(x) = (x² + 3x + 2)/(x + 1) for x ≠ −1. Find lim(x→−1) f(x).

Practice 15medium

If f(x) = (x³ - 8)/(x - 2) for x ≠ 2, then lim(x→2) f(x) equals:

Practice 16medium

Let f(x) = |x² - 4| for x ∈ ℝ. At which of the following points is f(x) NOT continuous?

Practice 17medium

Let f(x) = (x² + 1)/(x - 1). At which point(s) is f(x) discontinuous, and what is the nature of the discontinuity?

Practice 18medium

Evaluate lim(x→0) [sin(3x) - 3sin(x)] / x³.

Practice 19medium

For what value of k is the function f(x) = {(x² - 4)/(x - 2) if x ≠ 2; k if x = 2} continuous at x = 2?

Practice 20medium

Let f(x) = |x - 1|. Which of the following statements about f is correct?

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60-Second Revision — Limits & Continuity

  • Remember: lim(x→0) (sin ax)/(bx) = a/b - direct formula shortcut
  • Formula: f continuous at x=a means f(a) = lim(x→a) f(x)
  • Trap: 0/0 doesn't mean limit doesn't exist - always simplify first
  • Method: Factor polynomials when direct substitution fails
  • Standard: lim(x→0) (sin x)/x = 1 and lim(x→0) (1-cos x)/x² = 1/2
  • Check: For continuity, verify function value equals limit value
  • Quick tip: Use L'Hospital's rule only for 0/0 or ∞/∞ forms
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