#### Differential Calculus - Limits and Continuity Two Marks Questions

11th Standard

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Maths

Time : 00:45:00 Hrs
Total Marks : 20
10 x 2 = 20
1. 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
2. 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
3. Compute $lim_{x\rightarrow8}(5x)$

4. Compute$lim_{x\rightarrow-2}(-{3\over 2}x)$

5. Find the positive integer n so that $lim_{x\rightarrow 3}{x^n-3^n\over x-3}=27$

6. Evaluate $\lim _{ x\rightarrow a }{ \frac { \sqrt { x } +\sqrt { a } }{ x+a } }$

7. Evaluate$\lim _{ x\rightarrow 0 }{ \frac { \sqrt { 1+x } +\sqrt { 1-x } }{ 1+x } }$

8. $If\lim _{ x\rightarrow 2 }{ \frac { { x }^{ n }-{ 2 }^{ n } }{ x-2 } } =80\quad and\quad n\in N,\quad find\quad n.$

9. Evaluate$\lim _{ x\rightarrow 0 }{ \frac { { a }^{ x }-1 }{ { b }^{ x }-1 } }$

10. 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.