Calculus Quest
Interactive Practice · Limits & Continuity

Master Limits & Continuity with Maral.

Practice the exact concepts from Maral's lecture notes. Each question is designed to build your understanding of limits, trigonometric limits, radical limits, and continuity. Answer, check, and learn from detailed solutions.

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Interactive Limits & Continuity

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Question 1 Factoring
Evaluate:
limx→2 x² - 4x² - 5x + 6
Solution:
Substitute x = 2: (0)/(0) → Indeterminate.
Factor: x² - 4 = (x - 2)(x + 2)
x² - 5x + 6 = (x - 2)(x - 3)
Cancel (x - 2):
limx→2 x + 2x - 3 = 4-1 = -4
Question 2 Trig Limits
Evaluate:
limx→0 3x · tan(7x)sin²(5x)
Solution:
Use approximations: sin(5x) ≈ 5x, tan(7x) ≈ 7x.
limx→0 3x · 7x(5x)² = 21x²25x² = 21/25
Question 3 Radical Limits
Evaluate:
limx→-1⁺ 4(x - 1)√(x+1) - √(2x+2)
Solution:
Multiply by conjugate: (√(x+1) + √(2x+2))
Denominator becomes (x+1) - (2x+2) = -(x+1).
Factor √(x+1) from numerator.
Simplify to get -8 (as in notes).
Question 4 Continuity
For f(x) = { 0, x≤0; x, 0<x≤1; 2x-1, 1≤x<4; 2x, x≥4 },
where is f(x) discontinuous?
Solution:
Check x=0: L=0, R=0, f(0)=0 → Continuous.
Check x=1: L=1, R=1, f(1)=1 → Continuous.
Check x=4: L=7, R=8 → Not equal.
Discontinuous at x = 4.
Question 5 Trig Limits
Evaluate:
limx→0 sin(3x)x
Solution:
Use limu→0 sin(u)/u = 1.
Rewrite: 3 · sin(3x)/(3x).
As x→0, 3x→0, so sin(3x)/(3x) → 1.
Answer = 3.
Question 6 Trig Limits
Evaluate:
limx→0 tan(5x)2x
Solution:
Use tan(u) ≈ u as u→0.
tan(5x) ≈ 5x.
(5x)/(2x) = 5/2.
Answer = 5/2.
Question 7 Trig Limits
Evaluate:
limx→0 1 - cos(x)x²
Solution:
Use cos(x) ≈ 1 - x²/2.
1 - cos(x) ≈ x²/2.
(x²/2) / x² = 1/2.
Answer = 1/2.
Question 8 Factoring
Evaluate:
limx→1 x³ - 1x - 1
Solution:
Factor: x³ - 1 = (x - 1)(x² + x + 1).
Cancel (x - 1).
Substitute x = 1: 1² + 1 + 1 = 3.
Answer = 3.
Question 9 Radical Limits
Evaluate:
limx→0 √(x+4) - 2x
Solution:
Multiply by conjugate: (√(x+4) + 2).
Numerator: (x+4) - 4 = x.
Denominator: x(√(x+4) + 2).
Cancel x: 1/(√(x+4) + 2).
Substitute x=0: 1/(2+2) = 1/4.
Answer = 1/4.
Question 10 Trig Limits
Evaluate:
limx→0 sin(4x)sin(2x)
Solution:
Rewrite: (sin(4x)/(4x)) · (2x/sin(2x)) · (4x/2x).
As x→0, sin(4x)/(4x) → 1 and 2x/sin(2x) → 1.
Remaining: 4x/2x = 2.
Answer = 2.
Question 11 Continuity
Is f(x) = (x² - 1)/(x - 1) continuous at x = 1?
Solution:
Simplify: (x-1)(x+1)/(x-1) = x+1 for x ≠ 1.
Limit as x→1 is 2.
But f(1) is undefined (0/0).
So there is a removable discontinuity at x=1.
Question 12 Trig Limits
Evaluate:
limx→0 tan(x) - sin(x)x³
Solution:
Use expansions: tan(x) ≈ x + x³/3, sin(x) ≈ x - x³/6.
tan(x) - sin(x) ≈ x³/3 + x³/6 = x³/2.
(x³/2) / x³ = 1/2.
Answer = 1/2.
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What you'll practice

Each question targets a specific technique from Maral's lecture notes.

📐 Factoring Limits

Resolve 0/0 indeterminate forms by factoring polynomials and canceling common factors.

🎯 Trigonometric Limits

Use small-angle approximations and fundamental trig limits like sin(x)/x = 1.

🔬 Radical Limits

Rationalize numerators using conjugates to eliminate square roots and resolve 0/0 forms.

📊 Continuity

Check left-hand and right-hand limits at boundary points to determine continuity.