Class 11 • Maths • Chapter 12
LIMITS AND DERIVATIVES
True & False Quiz
Approach. Derive. Converge.
✓True
✗False
25
Questions
|
Ch.12
Chapter
|
XI
Class
Why True & False for LIMITS AND DERIVATIVES?
How this format sharpens your conceptual clarity
🔵 Limits and derivatives are the gateway to calculus — defining instantaneous rate of change, foundation of all physics.
✅ T/F tests the subtle but critical distinction: a limit can exist at a point where the function is undefined (TRUE — removable discontinuity).
🎯 lim(x→0) sin x/x = 1 (TRUE); lim(x→0)(1−cos x)/x = 0 (NOT 1 — common error).
📋
Read each statement carefully. Click True or False — instant feedback with explanation appears. Submit anytime; unattempted questions are marked Skipped.
Q 1
If \(f(x)=3x+5\), then \(\lim\limits_{x\to 2} f(x)=11\).
Q 2
If \(\lim\limits_{x\to a} f(x)\) exists, then \(f(a)\) must be defined.
Q 3
\(\lim\limits_{x\to 0} \sin x = 0\).
Q 4
If \(\lim\limits_{x\to a^-} f(x)\neq \lim\limits_{x\to a^+} f(x)\), then \(\lim\limits_{x\to a} f(x)\) does not exist.
Q 5
\(\lim\limits_{x\to 1} \dfrac{x^2-1}{x-1}=2\).
Q 6
\(\lim\limits_{x\to 0} \dfrac{\sin x}{x}=1\).
Q 7
\(\lim\limits_{x\to 0} \dfrac{1}{x}\) exists.
Q 8
If \(f(x)=|x|\), then \(\lim\limits_{x\to 0} f(x)=0\).
Q 9
\(\lim\limits_{x\to 0} \dfrac{|x|}{x}=1\).
Q 10
If \(f'(a)\) exists, then \(f(x)\) is continuous at \(x=a\).
Q 11
If a function is continuous at \(x=a\), then it must be differentiable at \(x=a\).
Q 12
The derivative of a constant function is zero.
Q 13
If \(f(x)=x^2\), then \(f'(1)=2\).
Q 14
\(\lim\limits_{x\to 0} \dfrac{e^x-1}{x}=1\).
Q 15
If \(f'(a)=0\), then \(f(x)\) has a maximum or minimum at \(x=a\).
Q 16
\(\lim\limits_{x\to \infty} \dfrac{1}{x}=0\).
Q 17
If \(f(x)=x^3\), then \(f'(0)=0\).
Q 18
\(\lim\limits_{x\to 0} \dfrac{\tan x}{x}=1\).
Q 19
If \(\lim\limits_{x\to a} f(x)=\infty\), then the limit does not exist.
Q 20
\(\lim\limits_{x\to 0} x\sin\left(\dfrac{1}{x}\right)=0\).
Q 21
If \(f(x)=\sqrt{x}\), then \(f'(0)\) exists.
Q 22
If \(\lim\limits_{x\to a} f(x)\) and \(\lim\limits_{x\to a} g(x)\) exist, then \(\lim\limits_{x\to a}[f(x)+g(x)]\) exists.
Q 23
\(\lim\limits_{x\to 0} \dfrac{\sin x - x}{x}=0\).
Q 24
If \(f'(x)\) exists for all real \(x\), then \(f(x)\) is continuous for all real \(x\).
Q 25
\(\lim\limits_{x\to 0} \dfrac{e^x - \cos x - x}{x^2}=\dfrac{1}{2}\).
Key Takeaways — LIMITS AND DERIVATIVES
Core facts for CBSE Boards & JEE
1
lim(x→0) sin x/x = 1 — fundamental trigonometric limit, memorise it.
2
lim(x→0)(1−cos x)/x = 0, NOT 1 — different from sin x/x.
3
A limit can exist at a point where f is not defined (removable discontinuity).
4
d/dx(xⁿ)=nxⁿ−¹; d/dx(sin x)=cos x; d/dx(cos x)=−sin x.
5
Product Rule: d/dx(uv)=u·v'+ v·u'; Quotient Rule: (vu'−uv')/v².
6
LHL = RHL is necessary AND sufficient for a limit to exist.