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Vidaara.orgClass 11 · Mathematics
CodeVID-M11-12-STD-01
Standard Limits — Assignment
Chapter: Limits and Derivatives
Topic: Standard Limits
Maximum Marks: 35
Time: 75 minutes
Name: ____________________ Roll No.: __________ Date: ____________

General Instructions

  • All questions are compulsory.
  • Section A carries 1 mark each, Section B 2 marks, Section C 3 marks and Section D 5 marks.
  • Show all working for Sections B, C and D. Only final answers are given at the end — for full solutions, raise your doubts with your teacher.
Section A — Multiple Choice Questions 5 × 1 = 5 marks
1.
$\displaystyle\lim_{x\to0}\dfrac{\sin x}{x}=$
  • A.$0$
  • B.$1$
  • C.$\infty$
  • D.$-1$
2.
$\displaystyle\lim_{x\to0}\dfrac{\tan x}{x}=$
  • A.$0$
  • B.$1$
  • C.$\infty$
  • D.$2$
3.
$\displaystyle\lim_{x\to a}\dfrac{x^n-a^n}{x-a}=$
  • A.$na^{n-1}$
  • B.$a^{n}$
  • C.$na^n$
  • D.$0$
4.
$\displaystyle\lim_{x\to0}\dfrac{1-\cos x}{x}=$
  • A.$1$
  • B.$0$
  • C.$\tfrac12$
  • D.$\infty$
5.
$\displaystyle\lim_{x\to0}\dfrac{e^x-1}{x}=$
  • A.$0$
  • B.$1$
  • C.$e$
  • D.$\infty$
Section B — Short Answer (2 marks) 4 × 2 = 8 marks
6.
Evaluate $\displaystyle\lim_{x\to0}\dfrac{\sin 2x}{x}$.
7.
Evaluate $\displaystyle\lim_{x\to0}\dfrac{\sin 5x}{\sin 3x}$.
8.
Evaluate $\displaystyle\lim_{x\to0}\dfrac{\tan x}{x}$.
9.
Evaluate $\displaystyle\lim_{x\to2}\dfrac{x^5-32}{x-2}$.
Section C — Short Answer (3 marks) 4 × 3 = 12 marks
10.
Evaluate $\displaystyle\lim_{x\to0}\dfrac{\sin 3x}{5x}$.
11.
Evaluate $\displaystyle\lim_{x\to0}\dfrac{1-\cos 2x}{x^2}$.
12.
Evaluate $\displaystyle\lim_{x\to0}\dfrac{\sin ax}{\sin bx}$.
13.
Evaluate $\displaystyle\lim_{x\to0}\dfrac{1-\cos x}{x^2}$.
Section D — Long Answer (5 marks) 2 × 5 = 10 marks
14.
Evaluate $\displaystyle\lim_{x\to0}\dfrac{\tan x-\sin x}{x^3}$.
15.
Evaluate $\displaystyle\lim_{x\to0}\dfrac{a^x-1}{x}$.

Answer Key

Section A — Multiple Choice Questions
  1. (B) $1$
  2. (B) $1$
  3. (A) $na^{n-1}$
  4. (B) $0$
  5. (B) $1$
Section B — Short Answer (2 marks)
  1. $2$.
  2. $\tfrac53$.
  3. $1$.
  4. $80$.
Section C — Short Answer (3 marks)
  1. $\tfrac35$.
  2. $2$.
  3. $\tfrac{a}{b}$.
  4. $\tfrac12$.
Section D — Long Answer (5 marks)
  1. $\tfrac12$.
  2. $\ln a$.
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