Q.1 If lim(x→2) (3x^2 - 4x + 1) = ?
5
3
1
7
Explanation - Substitute x = 2: 3(2)^2 - 4(2) + 1 = 12 - 8 + 1 = 5.
Correct answer is: 5
Q.2 Evaluate lim(x→0) (sin 5x)/x
0
1
5
∞
Explanation - lim(x→0) (sin 5x)/x = lim(x→0) (sin 5x)/(5x) * 5 = 1*5 = 5.
Correct answer is: 5
Q.3 The function f(x) = |x| is continuous at x = 0?
Yes
No
Only from right
Only from left
Explanation - Both left and right limits at x=0 are 0, and f(0)=0, so continuous.
Correct answer is: Yes
Q.4 lim(x→0) (1 - cos x)/x^2 equals?
1
0
1/2
2
Explanation - Using L'Hospital or formula: 1 - cos x ≈ x^2/2 as x→0. So limit = 1/2.
Correct answer is: 1/2
Q.5 Determine lim(x→∞) (5x^2 + 3x + 1)/(2x^2 - x + 4)
5/2
∞
0
2/5
Explanation - Divide numerator & denominator by x^2: limit = 5/2 as x→∞.
Correct answer is: 5/2
Q.6 The function f(x) = 1/x is continuous in which interval?
(-∞,0)
(0,∞)
(-∞,0) ∪ (0,∞)
All real numbers
Explanation - f(x) is undefined at x=0, but continuous elsewhere.
Correct answer is: (-∞,0) ∪ (0,∞)
Q.7 Evaluate lim(x→1) (x^3 - 1)/(x - 1)
1
2
3
0
Explanation - Factor numerator: (x-1)(x^2 + x + 1)/(x-1) = x^2 + x + 1. At x=1: 1+1+1=3.
Correct answer is: 3
Q.8 The limit lim(x→0) (e^x - 1)/x equals?
0
1
e
∞
Explanation - Using standard limit: lim(x→0) (e^x - 1)/x = 1.
Correct answer is: 1
Q.9 lim(x→0) ln(1+x)/x = ?
0
1
∞
-1
Explanation - Using standard limit: ln(1+x) ≈ x as x→0, so limit = 1.
Correct answer is: 1
Q.10 Check continuity: f(x) = {x^2, x≠2; 5, x=2}. Is f(x) continuous at x=2?
Yes
No
Only from left
Only from right
Explanation - lim(x→2) f(x) = 4 ≠ f(2) = 5, so not continuous at x=2.
Correct answer is: No
Q.11 lim(x→∞) (3x^3 - x)/(x^3 + 2) = ?
3
0
∞
1
Explanation - Divide numerator & denominator by x^3: limit = 3/1 = 3.
Correct answer is: 3
Q.12 The function f(x) = sin(1/x) is continuous at x=0?
Yes
No
Only from right
Only from left
Explanation - Limit does not exist as x→0; oscillates infinitely.
Correct answer is: No
Q.13 lim(x→0) (tan x)/x = ?
0
1
∞
-1
Explanation - Using standard limit: lim(x→0) (tan x)/x = 1.
Correct answer is: 1
Q.14 lim(x→0) (x - sin x)/x^3 = ?
1/6
0
1/2
∞
Explanation - Using Taylor series: sin x ≈ x - x^3/6, so limit = 1/6.
Correct answer is: 1/6
Q.15 Evaluate lim(x→0) (1 + x)^(1/x)
1
e
0
∞
Explanation - Standard limit: lim(x→0) (1+x)^(1/x) = e.
Correct answer is: e
Q.16 If lim(x→0) f(x)/x = 7, then lim(x→0) f(x) = ?
0
7
∞
Cannot be determined
Explanation - Since f(x)/x → 7, f(x) → 7x → 0 as x→0.
Correct answer is: 0
Q.17 Check continuity: f(x) = {x^2-1, x<1; 2, x=1; 1-x, x>1}. Is f(x) continuous at x=1?
Yes
No
Only from left
Only from right
Explanation - Left limit = 0, right limit = 0, f(1)=2; not continuous.
Correct answer is: No
Q.18 lim(x→0) (1 - e^(-x))/x = ?
0
1
-1
∞
Explanation - Standard limit: lim(x→0) (1 - e^(-x))/x = 1.
Correct answer is: 1
Q.19 lim(x→0) (x^2 + 2x)/(x) = ?
0
2
∞
x
Explanation - Divide numerator by x: (x + 2) → 2 as x→0.
Correct answer is: 2
Q.20 lim(x→∞) (7x^2 + x)/(x^2 - 3) = ?
7
0
∞
1/7
Explanation - Divide numerator & denominator by x^2: limit = 7/1 = 7.
Correct answer is: 7
Q.21 Check continuity: f(x) = {sin x / x, x≠0; 1, x=0}. Is f(x) continuous at x=0?
Yes
No
Only from left
Only from right
Explanation - lim(x→0) sin x / x = 1 = f(0), so continuous.
Correct answer is: Yes
Q.22 lim(x→0) (x - tan x)/x^3 = ?
-1/3
1/3
0
∞
Explanation - Using series: tan x ≈ x + x^3/3, so limit = -1/3.
Correct answer is: -1/3
Q.23 lim(x→2) (x^2 - 4)/(x - 2) = ?
0
2
4
∞
Explanation - Factor numerator: (x-2)(x+2)/(x-2) = x+2 → 4 at x=2.
Correct answer is: 4
Q.24 The function f(x) = 1/(x-1) is continuous in which interval?
(-∞,1)
(1,∞)
(-∞,1) ∪ (1,∞)
All real numbers
Explanation - f(x) undefined at x=1, continuous elsewhere.
Correct answer is: (-∞,1) ∪ (1,∞)
Q.25 lim(x→0) (x sin x)/(1 - cos x) = ?
0
2
1
∞
Explanation - Using limits: 1-cos x ≈ x^2/2, sin x ≈ x, so limit = (x*x)/(x^2/2)=2.
Correct answer is: 2