Limits and Continuity / सीमाएँ और सततता

Chapter Overview / अध्याय का अवलोकन

This chapter introduces the concept of limits of a function and the idea of continuity at a point. It forms the foundation of calculus.
यह अध्याय किसी फलन की सीमा (Limit) और किसी बिंदु पर उसकी सततता (Continuity) की अवधारणा से परिचित कराता है। यह कैलकुलस की नींव है।

Important Formulas / महत्वपूर्ण सूत्र

• lim(x→a) c = c
• lim(x→a) x = a
• lim(x→0) (sin x)/x = 1
• lim(x→0) (1 - cos x)/x² = 1/2
• Function f(x) is continuous at x = a if: lim(x→a⁻)f(x) = lim(x→a⁺)f(x) = f(a)

Hint-Based Questions (with Step-by-Step Hints) / संकेत-आधारित प्रश्न

1. Q1: Evaluate: lim(x→0) (sin x)/x
   
   Hint Step 1

   Recognize the standard limit form.
   Hint Step 2

   The standard limit lim(x→0)(sin x)/x = 1.
   Answer

   1
2. Q2: Evaluate: lim(x→0) (1 - cos x)/x²
   
   Hint Step 1

   This is a standard limit identity.
   Hint Step 2

   lim(x→0)(1 - cos x)/x² = 1/2
   Answer

   1/2
3. Q3: lim(x→2) (x² - 4)/(x - 2)
   
   Hint Step 1

   Factor the numerator: (x-2)(x+2)
   Hint Step 2

   Cancel (x-2); lim becomes lim(x→2)(x+2)
   Answer

   4
4. Q4: Check continuity of f(x) = x² at x = 3
   
   Hint Step 1

   Find left-hand limit, right-hand limit, and f(3)
   Hint Step 2

   All equal to 9, hence continuous
   Answer

   Yes, f(x) is continuous at x = 3
5. Q5: lim(x→1) (x² - 1)/(x - 1)
   
   Hint Step 1

   Factor numerator: (x - 1)(x + 1)
   Hint Step 2

   Cancel (x - 1); lim becomes lim(x→1)(x + 1)
   Answer

   2

Practice Questions (with Answers Only) / अभ्यास प्रश्न

1. lim(x→0) (tan x)/x = 1
2. lim(x→1) (x³ - 1)/(x - 1) = 3
3. Is f(x) = |x| continuous at x = 0? Yes
4. lim(x→3) (x² - 9)/(x - 3) = 6
5. lim(x→π/2) (cos x)/(π - 2x) = 0