#### Model Questions-Differential Calculus

11th Standard

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Maths

Time : 01:00:00 Hrs
Total Marks : 60

Part A

10 x 1 = 10
1. $lim_{x\rightarrow\infty}{sin \ x \over x}$

(a)

1

(b)

0

(c)

$\infty$

(d)

-$\infty$

2. $lim_{x\rightarrow {\pi/2}}{2x-\pi\over cosx}$

(a)

2

(b)

1

(c)

-2

(d)

0

3. $lim_{x \rightarrow \infty}{\sqrt{x^2-1}\over 2x+1}=$

(a)

1

(b)

0

(c)

-1

(d)

$1\over 2$

4. If f(x)=x(-1)$\left\lfloor 1\over x \right\rfloor$,$x\le0$,then the value of $lim_{x\rightarrow 0}f(x)$ is equal to

(a)

-1

(b)

0

(c)

2

(d)

4

5. $lim_{n \rightarrow \infty}({1\over n^2}+{2\over n^2}+{3\over n^2}+..+{n\over n^2})$ is

(a)

$1\over 2$

(b)

0

(c)

1

(d)

$\infty$

6. $lim_{x \rightarrow 0}{e^{tan \ x}-e^x\over tan x-x}=$

(a)

1

(b)

e

(c)

${1\over2}$

(d)

0

7. The value of $lim_{x \rightarrow 0}{sin x\over \sqrt{x^2}}$ is

(a)

1

(b)

-1

(c)

0

(d)

$\infty$

8. The value of $lim_{x\rightarrow k^-}x-\left\lfloor x \right\rfloor$where k is an integer is

(a)

-1

(b)

1

(c)

0

(d)

2

9. At x$={3\over 2}$ the function $f(x)={|2x-3|\over 2x-3}$ is

(a)

continuous

(b)

discontinuous

(c)

differentiable

(d)

non-zero

10. Let a function f be defined by $f(x)={x-|x|\over x}$ for x $\neq$0 and f(0)=2. Then f is

(a)

continuous nowhere

(b)

continuous everywhere

(c)

continuous for all x except x = 1

(d)

continuous for all x except x = 0

11. Part B

10 x 2 = 20
12. Consider the function f(x) = $\sqrt{x},x\ge0.$
Does$lim_{x\rightarrow0}f(x)$ exist?

13. Complete the table using calculator and use the result to estimate the limit.
$lim_{x\rightarrow 2}{x-2\over x^2-x-2}$

 x 1.9 1.99 1.999 2.001 2.01 2.1 f(x)
14. Complete the table using calculator and use the result to estimate the limit.
$lim_{x\rightarrow{-3}}{\sqrt{1-x}-2\over x+3}$

 x -3.1 -3.01 -3.00 -2.999 -2.99 -2.9 f(x)
15. Complete the table using calculator and use the result to estimate the limit.
$lim_{x\rightarrow0}{cos x-1\over x}$

 x -0.1 -0.01 -0.001 0.001 0.01 0.1 f(x)
16. Suppose that the diameter of an animal’s pupils is given by $f(x)={160x^{-0.4}+90\over 4x^{-0.4}+15},$ where x is the intensity of light and f(x) is in mm. Find the diameter of the pupils with minimum light.

17. Find the left and right limits of $f(x)={x^2-4\over (x^2+4x+4)(x+3)}at \ x=-2$ .

18. Evaluate the following limits :$lim_{x\rightarrow \infty}(1+{k\over x})^{m\over x}$

19. Determine if f defined by

20. Examine the continuity of the following: e2x + x2

21. Find the points of discontinuity of the function f, where f(x)={$\begin{matrix} { x }^{ 3 }-3, & if\quad x\le 2 \\ { x }^{ 2 }+1, & if\quad x>2 \end{matrix}$

22. Part C

5 x 3 = 15
23. Use the graph to find the limits (if it exists). If the limit does not exist, explain why?
$lim_{x\rightarrow3}(4-x)$.

24. Use the graph to find the limits (if it exists). If the limit does not exist, explain why?
$lim_{x\rightarrow1}(x^2+2)$

25. Use the graph to find the limits (if it exists). If the limit does not exist, explain why?
$lim_{x\rightarrow2}f(x)$

26. Use the graph to find the limits (if it exists). If the limit does not exist, explain why?
$lim_{x\rightarrow1}f(x)$

27. Use the graph to find the limits (if it exists). If the limit does not exist, explain why?
$lim_{x\rightarrow3}{1\over x-3}$

28. Part D

29. Sketch the graph of f, then identify the values of x0 for which $lim_{x\rightarrow{x_o}}f(x)$ exists.
30. Calculate $lim_{x\rightarrow0}{1\over (x^2+x^3)}$