Limits, Continuity and Differentiability

This chapter has

Limits, Continuity and Differentiability — solved practice questions

8 JEE Mathematics questions with step-by-step solutions. Attempt each, then reveal the answer.

1. Q1easy

   The value of the limit as x approaches 0 of (sin 5x)/(sin 3x) is:

   • A1
   • B3/5
   • C5/3
   • D15
   Show answer & solution

   Correct answer: (C) 5/3

   Writing (sin 5x)/(5x) · (3x)/(sin 3x) · (5/3) and using (sin t)/t -> 1 gives the limit 5/3.

2. Q2easy

   The value of the limit as x approaches 0 of (1 - cos 2x)/x^2 is:

   • A1
   • B2
   • C1/2
   • D4
   Show answer & solution

   Correct answer: (B) 2

   Using 1 - cos 2x = 2 sin^2 x, the expression is 2(sin x / x)^2 which tends to 2.

3. Q3easy

   If f(x) = (x^2 - 9)/(x - 3) for x not equal to 3 and f(3) = k, then f is continuous at x = 3 when k equals:

   • A0
   • B3
   • C9
   • D6
   Show answer & solution

   Correct answer: (D) 6

   For x not equal to 3, f(x) = x + 3, whose limit as x -> 3 is 6, so continuity requires k = 6.

4. Q4easy

   The value of the limit as x approaches 2 of (x^3 - 8)/(x - 2) is:

   • A12
   • B8
   • C6
   • D4
   Show answer & solution

   Correct answer: (A) 12

   This matches the standard form (x^n - a^n)/(x - a) -> n·a^(n-1) = 3·2^2 = 12; equivalently factor x^3 - 8 = (x - 2)(x^2 + 2x + 4).

5. Q5medium

   The value of the limit as x approaches infinity of (sqrt(x^2 + x) - x) is:

   • A0
   • B1/2
   • C1
   • Dinfinity
   Show answer & solution

   Correct answer: (B) 1/2

   Multiplying by the conjugate gives x/(sqrt(x^2 + x) + x), which tends to 1/2 as x -> infinity.

6. Q6medium

   The value of the limit as x approaches 0 of (tan x - sin x)/x^3 is:

   • A0
   • B1/6
   • C1/2
   • D1
   Show answer & solution

   Correct answer: (C) 1/2

   tan x - sin x = sin x (1 - cos x)/cos x. Using sin x ~ x, 1 - cos x ~ x^2/2, cos x -> 1 gives (x·x^2/2)/x^3 = 1/2.

7. Q7medium

   The value of the limit as x approaches 0 of (1 + 2x)^(1/x) is:

   • Ae
   • B1
   • C2e
   • De^2
   Show answer & solution

   Correct answer: (D) e^2

   This is the standard form (1 + kx)^(1/x) -> e^k with k = 2, giving e^2.

8. Q8medium

   The value of the limit as x approaches 0 of (x - sin x)/x^3 is:

   • A1/6
   • B1/2
   • C1/3
   • D0
   Show answer & solution

   Correct answer: (A) 1/6

   Using sin x = x - x^3/6 + ..., x - sin x = x^3/6 + higher terms, so the limit is 1/6.

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