#### Differential Calculus - Limits and Continuity Model Question Paper

11th Standard

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Maths

Time : 01:30:00 Hrs
Total Marks : 50
5 x 1 = 5
1. $lim_{x\rightarrow\infty}{sin \ x \over x}$

(a)

1

(b)

0

(c)

$\infty$

(d)

-$\infty$

2. 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

3. If $lim_{x \rightarrow 0}{sin \ px\over tan \ 3x}=4$ , then the value of p is

(a)

6

(b)

9

(c)

12

(d)

4

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

(a)

1

(b)

e

(c)

${1\over2}$

(d)

0

5. The function is not defined for x = −1. The value of f(−1) so that the function extended by this value is continuous is

(a)

${2\over3}$

(b)

-${2\over3}$

(c)

1

(d)

0

6. 8 x 2 = 16
7. Consider the function f(x) = $\sqrt{x},x\ge0.$
Does$lim_{x\rightarrow0}f(x)$ exist?

8. 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) 0.34482 0.33444 0.33344 0.333222 0.33222 0.332258
9. 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) – 0.24845 – 0.24984 – 0.24998 – 0.25001 – 0.25015 – 0.25158
10. Compute $lim_{x\rightarrow1}{x^3-1\over x-1}$

11. $f(x)=tan \ x \ at \ x={\pi\over 2}.$

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

13. Examine the continuity of the following: x + sin x

14. Examine the continuity of the following:cot x + tan x

15. 3 x 3 = 9
16. Calculate $\lim _{ x\rightarrow0}{|x| }$.

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

18. Use the graph to find the limits (if it exists). If the limit does not exist, explain why?
$lim_{x\rightarrow5}{|x-5|\over x-5}$

19. 4 x 5 = 20
20. Check if $lim_{x\rightarrow-58}f(x)$exists or not, where

21. Write a brief description of the meaning of the notation $lim_{x\rightarrow8}f(x)=25.$

22. Evaluate the following limits :
$lim_{x\rightarrow5}{\sqrt{x-1}-2\over x-5}$

23. A tomato wholesaler finds that the price of a newly harvested tomatoes is Rs. 0.16 per kg if he purchases fewer than 100 kgs each day. However, if he purchases at least 100 kgs daily, the price drops to Rs. 0.14 per kg. Find the total cost function and discuss the cost when the purchase is 100 kgs.