x y z (x,y,z) d=√(Δx²+Δy²+Δz²)
lim
Chapter 12  ·  Class XI Mathematics  ·  MCQ Practice

MCQ Practice Arena

Limits & Derivatives

Where Calculus Begins — Speed Through Limits, Ace the Derivative

📋 50 MCQs ⭐ 0 PYQs ⏱ 75 sec/Q
🚀 Start Practising 📖 Revise Notes First

MCQ Bank Snapshot

50Total MCQs
18Easy
20Medium
12Hard
0PYQs
75 secAvg Time/Q
12Topics
Easy 36% Medium 40% Hard 24%

Why Practise These MCQs?

JEE MainJEE AdvancedCBSEBITSATKVPY

Limits & Derivatives opens the calculus sequence — the dominant topic of JEE. JEE Main tests standard limits, 0/0 form simplification, and derivative rules in 4–5 MCQs. KVPY has elegant limit evaluation problems. CBSE places first-principles derivative proofs as MCQs. Mastery here guarantees success in Class XII Calculus.

Topic-wise MCQ Breakdown

Concept of Limit5 Q
Algebra of Limits6 Q
Algebraic Limits (0/0)5 Q
Trigonometric Limits9 Q
Exponential/Log Limits4 Q
Limits at Infinity2 Q
First Principles Deriv.1 Q
Derivative — Polynomials6 Q
Derivative — Trig Fns4 Q
Product Rule1 Q
Chain Rule1 Q
Conceptual Derivatives6 Q

Must-Know Formulae Before You Start

Recall these cold before attempting MCQs — they appear in >70% of questions.

$\lim\limits_{(x\to a)} (xⁿ−aⁿ)/(x−a) = naⁿ⁻¹$
$\lim\limits_{(x\to 0)} \sin x/x = 1$
$\lim\limits_{(x\to 0)} (eˣ−1)/x = 1$
$d/dx(\sin x)=\cos x, d/dx(xⁿ)=nxⁿ⁻¹$
$(uv)'=u'v+uv' \text{(Product Rule)}$

MCQ Solving Strategy

For 0/0 form algebraic limits, factorise the numerator and cancel (x−a). For trig limits, always convert to standard sinx/x or tanx/x form by multiplying and dividing. For derivative MCQs, the product rule is the most used — write it as (uv)' = u'v + uv' and identify u and v clearly before differentiating. First-principles MCQs: use f(x+h)−f(x)/h and let h→0.

⚠ Common Traps & Errors

Difficulty Ladder

Work through each rung in order — do not jump to Hard before mastering Easy.

① Easy

Substitute directly, standard limit values, basic derivative rules

② Medium

Factorise for 0/0 form, product and quotient rule, trig limits

③ Hard

Nested limits, limits at infinity, first-principles for complex functions

★ PYQ

JEE Main — algebraic + trig limits; KVPY — elegant conceptual limits

Continue Your Preparation

📖 Chapter Notes Revise concepts & theory 🧠 MCQ Practice Topic-wise Questions 🎯 PYQ Bank Previous year questions ✔️❌ True / False Concept-based True & False questions 📘✔️ Exercises Step by step Solutions to textbook exercises ➡ Next Chapter Mathematical Reasoning MCQs
🎯 Knowledge Check

Maths — LIMITS AND DERIVATIVES

50 Questions Class 11 MCQs
1
Evaluate \(\lim\limits_{x\to 2}(x+3)\).
(Basic Concept)
2
Find \(\lim\limits_{x\to 1}(2x^2-3x+1)\).
(Basic Concept)
3
Evaluate \(\lim\limits_{x\to 0}(5x)\).
(Basic Concept)
4
Find \(\lim\limits_{x\to -1}(x^2+2x+1)\).
(Basic Concept)
5
Evaluate \(\lim\limits_{x\to 3}(x^2-9)\).
(Basic Concept)
6
Find \(\lim\limits_{x\to 1}\frac{x^2-1}{x-1}\).
(Factorisation)
7
Evaluate \(\lim\limits_{x\to 2}\frac{x^2-4}{x-2}\).
(Factorisation)
8
Find \(\lim\limits_{x\to 0}\frac{\sin x}{x}\).
(Standard Limit)
9
Evaluate \(\lim\limits_{x\to 0}\frac{\tan x}{x}\).
(Standard Limit)
10
Find \(\lim\limits_{x\to 0}\frac{1-\cos x}{x^2}\).
(Standard Limit)
11
Evaluate \(\lim\limits_{x\to 0}\frac{\sin 3x}{x}\).
(Standard Limit)
12
Find \(\lim\limits_{x\to 0}\frac{\sin 5x}{\sin 2x}\).
(Standard Limit)
13
Evaluate \(\lim\limits_{x\to 1}\frac{x^2+x-2}{x-1}\).
(Factorisation)
14
Find \(\lim\limits_{x\to 0}\frac{e^x-1}{x}\).
(Standard Result)
15
Evaluate \(\lim\limits_{x\to 0}\frac{\ln(1+x)}{x}\).
(Standard Result)
16
Find the derivative of \(f(x)=x^2\) at \(x=1\).
(Derivative from First Principle)
17
The derivative of a constant function is:
(Conceptual)
18
Find \(\frac{d}{dx}(3x)\).
(Basic Derivative)
19
Evaluate \(\frac{d}{dx}(x^3)\).
(Basic Derivative)
20
Find the derivative of \(x^2+5x+1\).
(Basic Derivative)
21
The geometrical meaning of derivative at a point is:
(Conceptual)
22
If \(y=x^n\), then \(\frac{dy}{dx}\) equals:
(Formula Based)
23
Find \(\frac{d}{dx}(\sqrt{x})\).
(Intermediate)
24
Evaluate \(\lim\limits_{x\to 0}\frac{\sin x - x}{x^3}\).
(Advanced Limit)
25
Find \(\frac{d}{dx}(x^2\sin x)\).
(Product Rule – Intro)
26
Evaluate \(\lim\limits_{x\to 0}\frac{e^x-\cos x}{x}\).
(Advanced Limit)
27
Find the derivative of \(\sin x\).
(Standard Derivative)
28
The derivative of \(\cos x\) is:
(Standard Derivative)
29
Find \(\frac{d}{dx}(\tan x)\).
(Standard Derivative)
30
Evaluate \(\lim\limits_{x\to a}\frac{x^2-a^2}{x-a}\).
(Algebraic Limit)
31
If \(f'(x)=0\) for all \(x\), then \(f(x)\) is:
(Conceptual)
32
Find \(\frac{d}{dx}(1/x)\).
(Intermediate)
33
Evaluate \(\lim\limits_{x\to 0}\frac{\tan x - x}{x^3}\).
(Advanced Limit)
34
The derivative represents:
(Conceptual)
35
Find \(\frac{d}{dx}(x^4-3x^2)\).
(Intermediate)
36
Evaluate \(\lim\limits_{x\to 0}\frac{\sin x}{x}\cdot\frac{1}{\cos x}\).
(Combination of Limits)
37
Find the derivative of \(2x^3+5\).
(Intermediate)
38
If \(y=x^2\), then \(\frac{dy}{dx}\) at \(x=0\) is:
(Conceptual)
39
Evaluate \(\lim\limits_{x\to 0}\frac{1}{x}\).
(One-sided Concept)
40
Find \(\frac{d}{dx}(\ln x)\).
(Standard Derivative)
41
The derivative of \(e^x\) is:
(Standard Derivative)
42
Evaluate \(\lim\limits_{x\to 0}\frac{\sqrt{1+x}-1}{x}\).
(Advanced Limit)
43
Find \(\frac{d}{dx}(x^{-2})\).
(Intermediate)
44
If \(f(x)=x^3\), then \(f'(2)\) equals:
(Application)
45
Evaluate \(\lim\limits_{x\to 0}\frac{e^{2x}-1}{x}\).
(Advanced Limit)
46
The slope of the tangent at \(x=a\) is given by:
(Conceptual)
47
Find \(\frac{d}{dx}(x^2+1)^2\).
(Chain Rule – Intro)
48
Evaluate \(\lim\limits_{x\to 0}\frac{\sin 2x}{x}\).
(Standard Limit)
49
If \(y=x^n\), the derivative at \(x=1\) is:
(Application)
50
The limit \(\lim\limits_{x\to 0^+}\ln x\) is:
(Higher Difficulty)
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Frequently Asked Questions

A limit describes the value that a function \(f(x)\) approaches as \(x\) approaches a particular number, written as \(\lim_{x\to a} f(x)\).

It means that the values of \(f(x)\) can be made arbitrarily close to \(L\) by taking \(x\) sufficiently close to \(a\), but not necessarily equal to \(a\).

No, the limit depends on the behavior of the function near the point, not necessarily on the value of \(f(a)\).

The left-hand limit is \(\lim_{x\to a^-} f(x)\), where \(x\) approaches \(a\) from values less than \(a\).

The right-hand limit is \(\lim_{x\to a^+} f(x)\), where \(x\) approaches \(a\) from values greater than \(a\).

A limit exists at \(x=a\) if both left-hand and right-hand limits exist and are equal.

An infinite limit occurs when \(f(x)\) increases or decreases without bound as \(x\) approaches a value, written as \(\lim_{x\to a} f(x)=\infty\).

For a constant function \(f(x)=c\), \(\lim_{x\to a} c = c\) for any real number \(a\).

For \(f(x)=x\), \(\lim_{x\to a} x = a\).

If \(\lim_{x\to a} f(x)=L\) and \(\lim_{x\to a} g(x)=M\), then \(\lim_{x\to a} [f(x)+g(x)]=L+M\).

\(\lim_{x\to a} [f(x)-g(x)] = L-M\), provided the individual limits exist.

For a constant \(k\), \(\lim_{x\to a} kf(x)=k\lim_{x\to a} f(x)=kL\).

\(\lim_{x\to a} [f(x)g(x)] = LM\), if both limits exist.

\(\lim_{x\to a} \frac{f(x)}{g(x)}=\frac{L}{M}\), provided \(M\neq 0\).

The limit of a polynomial at \(x=a\) is found by direct substitution of \(x=a\).

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