Q.1 What is the value of lim(x→0) (sin x / x)?
0
1
∞
Does not exist
Explanation - The limit of sin(x)/x as x approaches 0 is a standard trigonometric limit equal to 1.
Correct answer is: 1
Q.2 lim(x→∞) (1/x) equals?
0
1
∞
Does not exist
Explanation - As x approaches infinity, 1/x approaches 0.
Correct answer is: 0
Q.3 Evaluate lim(x→0) (1 - cos x) / x².
0
1/2
1
Does not exist
Explanation - Using the expansion cos x ≈ 1 - x²/2, the numerator becomes x²/2, so the limit is 1/2.
Correct answer is: 1/2
Q.4 What is lim(x→2) (x² - 4)/(x - 2)?
0
2
4
Does not exist
Explanation - Factoring gives (x+2)(x-2)/(x-2). Cancel terms to get x+2 → 4.
Correct answer is: 4
Q.5 lim(x→0⁺) ln(x) equals?
0
1
-∞
∞
Explanation - As x approaches 0 from the right, ln(x) decreases without bound.
Correct answer is: -∞
Q.6 Evaluate lim(x→∞) (2x² + 3x)/(x² - x).
0
2
∞
Does not exist
Explanation - Divide numerator and denominator by x², leading terms dominate giving 2/1 = 2.
Correct answer is: 2
Q.7 Does lim(x→0) (|x|/x) exist?
Yes, 1
Yes, 0
No
Yes, -1
Explanation - Left-hand limit = -1, right-hand limit = 1. Since they differ, the limit does not exist.
Correct answer is: No
Q.8 Find lim(x→π/2⁻) tan(x).
0
1
∞
-∞
Explanation - As x approaches π/2 from the left, tan(x) increases without bound.
Correct answer is: ∞
Q.9 Evaluate lim(x→0) (e^x - 1)/x.
0
1
∞
Does not exist
Explanation - Derivative of e^x at 0 is 1, so the limit equals 1.
Correct answer is: 1
Q.10 lim(x→∞) (ln x / x) is?
0
1
∞
Does not exist
Explanation - x grows faster than ln x, so the fraction approaches 0.
Correct answer is: 0
Q.11 What is the left-hand limit of f(x) = 1/x as x → 0⁻?
0
∞
-∞
Does not exist
Explanation - As x approaches 0 from the left, 1/x decreases without bound.
Correct answer is: -∞
Q.12 Find lim(x→3) (x² - 9)/(x - 3).
3
6
9
Does not exist
Explanation - Factor as (x-3)(x+3)/(x-3). Cancel to get x+3 → 6.
Correct answer is: 6
Q.13 lim(x→0) (tan x / x) equals?
0
1
∞
Does not exist
Explanation - Using expansion tan x ≈ x near 0, ratio → 1.
Correct answer is: 1
Q.14 Evaluate lim(x→∞) (x / √(x²+1)).
0
1
∞
Does not exist
Explanation - Divide numerator and denominator by x, expression → 1.
Correct answer is: 1
Q.15 What is lim(x→0) (sin 2x / x)?
0
1
2
Does not exist
Explanation - Rewrite as (sin 2x)/(2x) × 2 → 1×2 = 2.
Correct answer is: 2
Q.16 lim(x→∞) (1 + 1/x)^x equals?
0
1
e
∞
Explanation - This is the definition of Euler’s number e.
Correct answer is: e
Q.17 Does lim(x→0) (x/|x|) exist?
Yes, 1
Yes, -1
Yes, 0
No
Explanation - Left-hand limit = -1, right-hand limit = 1. They are not equal, so limit does not exist.
Correct answer is: No
Q.18 Evaluate lim(x→∞) (√(x²+2x) - x).
0
1
∞
Does not exist
Explanation - Rationalize: (√(x²+2x) - x)(√(x²+2x)+x)/(√(x²+2x)+x) = 2x/(√(x²+2x)+x) → 1.
Correct answer is: 1
Q.19 lim(x→0) (a^x - 1)/x equals?
0
ln a
a
Does not exist
Explanation - Derivative of a^x at 0 is ln(a).
Correct answer is: ln a
Q.20 Find lim(x→0⁺) (1/x).
0
1
∞
-∞
Explanation - As x approaches 0 from the right, 1/x grows without bound.
Correct answer is: ∞
Q.21 Evaluate lim(x→0) (x² / sin²x).
0
1
∞
Does not exist
Explanation - Since sin x ≈ x, ratio (x²)/(x²) → 1.
Correct answer is: 1
Q.22 What is lim(x→∞) (x² / e^x)?
0
1
∞
Does not exist
Explanation - Exponential functions grow faster than polynomials, so limit → 0.
Correct answer is: 0
Q.23 Does lim(x→0) (sin(1/x)) exist?
Yes, 0
Yes, 1
Yes, -1
No
Explanation - sin(1/x) oscillates infinitely near 0, so the limit does not exist.
Correct answer is: No
Q.24 Evaluate lim(x→π) (sin x)/(x - π).
0
1
-1
Does not exist
Explanation - As x → π, numerator → 0 while denominator → 0, but sin x is 0 exactly at π. So overall limit = 0.
Correct answer is: 0
Q.25 What is lim(x→∞) (x^(1/x))?
0
1
∞
Does not exist
Explanation - Taking ln: (1/x)ln x → 0, so original limit = e^0 = 1.
Correct answer is: 1