11th Standard Syllabus & Materials
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TN 11th Tamil பீடு பெற நில் - துணைப்பாடம் - வாடிவாசல் Important Questions And Answers Study Material - QB365 Set A
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Published on: 21/09/2019
Differential Calculus - Limits and Continuity
Download Tamil Nadu 11th Standard Maths question papers, model tests, one-mark questions, important questions, and public exam papers in PDF format. Free study materials and answer keys for TN State Board students.
Questions + Answers key
Take MCQ Maths Test1.
For what value of k is the function \(f\left( x \right) =\begin{cases} \frac { \sin { 5x } }{ 3x } \quad if\quad x\neq 0 \\ k,\quad \quad \quad if\quad x=0 \end{cases}\) is continuous at x = 0.
2.
Evaluate\(\lim _{ x\rightarrow 0 }{ \frac { { a }^{ x }-1 }{ { b }^{ x }-1 } } \)
3.
\(If\lim _{ x\rightarrow 2 }{ \frac { { x }^{ n }-{ 2 }^{ n } }{ x-2 } } =80\quad and\quad n\in N,\quad find\quad n.\)
4.
Evaluate\(\lim _{ x\rightarrow 0 }{ \frac { \sqrt { 1+x } +\sqrt { 1-x } }{ 1+x } } \)
5.
Evaluate \(\lim _{ x\rightarrow a }{ \frac { \sqrt { x } +\sqrt { a } }{ x+a } } \)
6.
Compute\(lim_{x\rightarrow-2}(-{3\over 2}x)\)
7.
Compute \(lim_{x\rightarrow8}(5x)\)
8.
In problem, using the table estimate the value of the limit
\(lim_{x\rightarrow 0}{sin x\over x}\)
| x | -0.1 | -0.01 | -0.001 | 0.001 | 0.01 | 0.1 |
| f(x) | 0.99833 | 0.99998 | 0.99999 | 0.99999 | 0.99998 | 0.99833 |
9.
In problem, using the table estimate the value of the limit
\(lim_{x\rightarrow{0}}{\sqrt{x+3}-\sqrt{3}\over x}\)
| x | -0.1 | -0.01 | -0.001 | 0.001 | 0.01 | 0.1 |
| f(x) | 0.2911 | 0.2891 | 0.2886 | 0.2886 | 0.2885 | 0.28631 |
10.
Find the positive integer n so that \(lim_{x\rightarrow 3}{x^n-3^n\over x-3}=27\)
1.
Given \(f\left( x \right) =\begin{cases} \frac { \sin { 5x } }{ 3x } \quad if\quad x\neq 0 \\ k,\quad \quad \quad if\quad x=0 \end{cases}\)
\(\lim _{ x\rightarrow 0 }{ f\left( x \right) } =\lim _{ x\rightarrow 0 }{ \frac { \sin { 5x } }{ 3x } } =\lim _{ x\rightarrow 0 }{ \left( \frac { \sin { 5x } }{ 3x } \right) \times \left( \frac { 5 }{ 3 } \right) } =1\times \frac { 5 }{ 3 } =\frac { 5 }{ 3 } \)
Given f is continuous at x = 0
\(\lim _{ x\rightarrow 0 }{ f\left( x \right) } =f\left( 0 \right) =\frac { 5 }{ 3 } \Rightarrow k=\frac { 5 }{ 3 } \)
\( \therefore \quad k=\frac { 5 }{ 3 } .\)
2.
\(\lim _{ x\rightarrow 0 }{ \frac { { a }^{ x }-1 }{ { b }^{ x }-1 } } =\lim _{ x\rightarrow 0 }{ \frac { \frac { { a }^{ x }-1 }{ x } }{ \frac { { b }^{ x }-1 }{ x } } } =\frac { \lim _{ x\rightarrow 0 }{ \left( \frac { { a }^{ x }-1 }{ x } \right) } }{ \lim _{ x\rightarrow 0 }{ \left( \frac { { b }^{ x }-1 }{ x } \right) } } =\frac { \log { a } }{ \log { b } } \)
3.
\(Given,\quad \lim _{ x\rightarrow 2 }{ \frac { { x }^{ n }-{ 2 }^{ n } }{ x-2 } } =80\)
\( \Rightarrow n.{ 2 }^{ n-1 }=80\)
By trial method, put n 5
\(\Rightarrow 5({ 2 }^{ 5-1 })=80\)
\(\Rightarrow 5({ 2 }^{ 4 })=80\)
\( \Rightarrow 5(16)=80\)
\( \Rightarrow 80=80\)
\(\therefore n=5\)
4.
\(\lim _{ x\rightarrow 0 }{ \frac { \sqrt { 1+x } +\sqrt { 1-x } }{ 1+x } } =\frac { \sqrt { 1+0 } +\sqrt { 1-0 } }{ 1+0 } =\frac { \sqrt { 1 } +\sqrt { 1 } }{ 1 } =\frac { 2 }{ 1 } =2\)
5.
\(\lim _{ x\rightarrow a }{ \frac { \sqrt { x } +\sqrt { a } }{ x+a } } =\frac { \sqrt { a } +\sqrt { a } }{ a+a } =\frac { 2\sqrt { a } }{ 2a } =\frac { 1 }{ \sqrt { a } } \)
6.
\(lim_{x\rightarrow-2}(-{3\over 2}x)=-{3\over2}lim_{x\rightarrow-2}(x)=({-{3\over2}})(-2)=3.\)
7.
\(lim_{x\rightarrow8}(5x)=5lim_{x\rightarrow8}(x)=5\times8=40.\)
8.
Let \( f(x)=\frac{\sin x}{x}
\)
\(\therefore \lim _{x \rightarrow 0} \frac{\sin x}{x}=0.999 \ldots=1
\)
9.
Let \(
f(x) =\frac{\sqrt{x+3}-\sqrt{3}}{x}
\)
\(\therefore \lim _{x \rightarrow 0} \frac{\sqrt{x+3}-\sqrt{3}}{x} =0.288
\)
10.
\(lim_{x\rightarrow 3}{x^n-3^n\over x-3}=n.3^{n-1}=27\)
That is \(n.3^{n-1}=3\times3^2=3\times3^{3-1}\Rightarrow n=3\) .
11th Standard Syllabus & Materials
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TN 11th Tamil பீடு பெற நில் - செய்யுள் - காவடிச்சிந்து Important Questions And Answers Study Material - QB365 Set A
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Tamilnadu Stateboard 11th Standard Subjects
Maths
Commerce
Economics
Biology
Business Maths and Statistics
Accountancy
Computer Science
Physics
Chemistry
Maths
Biology
Economics
Physics
Chemistry
History
Business Maths and Statistics
Computer Science
Accountancy
Computer Applications
History
Computer Technology
Commerce
Computer Applications
Computer Technology
Tamil
English
French
Tamilnadu Stateboard Standards
