Class XI Assignment

1. LIMITS AND DERIVATIVES
1. Evaluate the left hand limit of the function: 𝑓( 𝑥) = {
| 𝑥−4|
𝑥−4
, 𝑥 ≠ 4
0, 𝑥 = 4
at x = 4.
2. Evaluate the left hand and right hand limits of the function defined by
f(x) = {
1 + 𝑥2, 𝑖𝑓 0 ≤ 𝑥 ≤ 1
2 − 𝑥, 𝑖𝑓 𝑥 > 1
at x = 1.
3. If f(x) = {
𝑥−| 𝑥|
𝑥
, 𝑥 ≠ 0
2, 𝑥 = 0
show that lim
𝑥→0
𝑓(𝑥) does not exit
4. If f(x) = {
5𝑥 − 4 0 < 𝑥 ≤ 1
4𝑥3 − 3𝑥 1 < 𝑥 < 2
, show that lim
𝑥→0
𝑓(𝑥) exit.
5. Let f(x) = {
𝑐𝑜𝑠 𝑥, 𝑖𝑓 𝑥 ≥ 0
𝑥 + 𝑘, 𝑖𝑓 𝑥 < 0
. Find the value of constant k, given that lim
𝑥→0
𝑓(𝑥) exit.
6. Let f(x) be a function by f(x) = {
4𝑥 − 5, 𝑖𝑓 𝑥 ≥ 2
𝑥 + 𝜆, 𝑖𝑓 𝑥 > 2
. find λ, if lim
𝑥→2
𝑓(𝑥) exit.
7. If f(x) = {
𝑚𝑥2 + 𝑛, 𝑥 < 0
𝑛𝑥 + 𝑚, 0 ≤ 𝑥 ≤ 1
𝑛𝑥3 + 𝑚, 𝑥 > 1
. For what values of integers m, n does the limits lim
𝑥→0
𝑓(𝑥) and lim
𝑥→1
𝑓(𝑥) exist.
8. Suppose f(x) = {
𝑎 + 𝑏𝑥 , 𝑥 < 1
4 , 𝑥 = 1
𝑏 − 𝑎𝑥 , 𝑥 > 1
and if lim
𝑥→1
𝑓(𝑥) = f(1). What are possible values of a and b?
9. Let f(x) = {
𝑥 + 1, 𝑖𝑓 𝑥 ≥ 0
𝑥 − 1, 𝑖𝑓 𝑥 < 0
, Prove that lim
𝑥→0
𝑓(𝑥) does not exit .
10. If f(x) = {
2𝑥 + 3 , 𝑥 ≤ 0
3(𝑥 + 1) , 𝑥 > 0
. find lim
𝑥→0
𝑓(𝑥) and lim
𝑥→1
𝑓(𝑥) .
11. Evaluate : lim
𝑥→2
𝑥2−5𝑥+6
𝑥2−4
.
12. Evaluate : lim
𝑥→2
𝑥3−3𝑥2+4
𝑥4−8𝑥2+16
.
13. Evaluate : lim
𝑥→2
𝑥3−6𝑥2+11𝑥−6
𝑥2−6𝑥 +8
.
14. Evaluate the following limits :
(i) lim
𝑥→3
𝑥4−81
𝑥2−9
.
(ii) lim
𝑥→2
(
1
𝑥−2
−
2
𝑥2−2𝑥
) .
(iii) lim
𝑥→0
(𝑎+𝑥)2−𝑎2
𝑥
.
(iv) lim
𝑥→1
√𝑥2−1+ √ 𝑥−1
√𝑥2−41
, x>1
15. Evaluate : lim
𝑥→0
√2+𝑥 −√2
𝑥
.
16. Evaluate : lim
𝑥→0
√𝑎2+𝑥2− √𝑎2−𝑥2
𝑥2
.
17. Evaluate : lim
𝑥→𝑎
√ 𝑎+2𝑥− √3𝑥
√3𝑎+𝑥− 2√ 𝑥
.

2. 18. Evaluate : lim
𝑥→4
3− √5+𝑥
1− √5−𝑥
.
19. Evaluate : lim
𝑥→2
𝑥10−1024
𝑥−2
.
20. Evaluate : lim
𝑥→9
𝑥3/2−27
𝑥−9
.
21. Evaluate : lim
𝑥→𝑎
𝑥 𝑚− 𝑎 𝑚
𝑥 𝑛− 𝑎 𝑛
.
22. Evaluate : lim
𝑥→𝑎
(𝑥+2)5/3 − (𝑎+2)5/3
𝑥−𝑎
.
23. Evaluate : lim
𝑥→0
(1−𝑥) 𝑛− 1
𝑥
.
24. Evaluate the following limits :
(i) lim
𝑥→0
1−𝑐𝑜𝑠 𝑚𝑥
1−𝑐𝑜𝑠𝑥
(ii) lim
𝑥→0
𝑡𝑎𝑛 𝑥−𝑠𝑖𝑛𝑥
𝑥3
(iii) lim
𝑥→0
𝑡𝑎𝑛 𝑥−𝑠𝑖𝑛 𝑥
𝑠𝑖𝑛3 𝑥
(iv) lim
𝑦→0
( 𝑥+𝑦) 𝑠𝑒𝑐( 𝑥+𝑦)− 𝑥 𝑠𝑒𝑐 𝑥
𝑦
(v) lim
𝑥→0
𝑠𝑒𝑐 4𝑥−𝑠𝑒𝑐 2𝑥
𝑠𝑒𝑐 3𝑥−𝑠𝑒𝑐 𝑥
(vi) lim
𝑥→0
1−𝑐𝑜𝑠 𝑥 √ 𝑐𝑜𝑠2𝑥
𝑥2
.
(vii) lim
𝑥→0
3𝑠𝑖𝑛 𝑥−4 𝑠𝑖𝑛3 𝑥
𝑥
(viii) lim
𝑥→0
𝑐𝑜𝑠 3𝑥−𝑐𝑜𝑠7𝑥
𝑥2
(ix) lim
𝑥→0
1−𝑐𝑜𝑠2 𝑥− 𝑡𝑎𝑛2 𝑥
𝑥 𝑠𝑖𝑛 𝑥
(x) lim
𝑥→0
√2− √1+𝑐𝑜𝑠 𝑥
𝑥2
(xi) lim
𝑥→0
1 –𝑐𝑜𝑠 4𝑥
𝑥2
(xii) lim
𝑥→
𝜋
4
𝑠𝑖𝑛 𝑥−𝑐𝑜𝑠 𝑥
𝑥−
𝜋
4
(xiii) lim
𝑥→
𝜋
6
√2𝑠𝑖𝑛 𝑥−𝑐𝑜𝑠 𝑥
𝑥−
𝜋
6
(xiv) lim
𝑥→
𝜋
2
𝑐𝑜𝑠 𝑥
𝜋
2
− 𝑥
(xv) lim
𝑥→
𝜋
6
√2𝑠𝑖𝑛 𝑥−𝑐𝑜𝑠 𝑥
𝑥−
𝜋
6
(xvi) lim
𝑥→
𝜋
2
𝑡𝑎𝑛 2 𝑥
𝑥−
𝜋
2
(xvii) lim
𝑥→0
𝑒−𝑥−1
𝑥
(xviii) lim
𝑥→0
𝑒 𝑥−𝑒−𝑥
𝑥

3. (xix) lim
𝑥→0
𝑙𝑜𝑔|1+ 𝑥3|
𝑠𝑖𝑛3 𝑥
(xx) lim
𝑥→0
𝑒 𝑥+2−𝑒2
𝑥
(xxi) lim
𝑥→0
𝑒3+𝑥− 𝑠𝑖𝑛 𝑥− 𝑒3
𝑥
(xxii) lim
𝑥→0
𝑒3𝑥−𝑒2𝑥
𝑥
(xxiii) lim
𝑥→0
𝑒 𝑡𝑎𝑛 𝑥−1
𝑡𝑎𝑛 𝑥
25. Differentiate each of the following by first principal :
(i)
𝑥2+ 1
𝑥
(ii)
2𝑥 +3
𝑥−2
(iii) 𝑒3𝑥
(iv) xsin x
(v) x cos x
(vi) tan 2
x
(vii) tan 2x
(viii) √ 𝑡𝑎𝑛 𝑥
(ix) cos (𝑥 −
𝜋
8
)
26. If y = √
1−𝑐𝑜𝑠 2𝑥
1+𝑐𝑜𝑠 2𝑥
then find
𝑑𝑦
𝑑𝑥
27. Differentiate the following functions w. r. t. s. x :
(i) x4
- 2 sin x + 3 cos x
(ii) 3x
+ x3
+ 33
(iii)
𝑥3
3
− 2 √ 𝑥 +
5
𝑥2
(iv) (√ 𝑥 +
1
√ 𝑥
)
3
(v)
𝑥2+ 3𝑥+4
𝑥
(vi) cos (x + a)
(vii)
𝑒 𝑥
1+𝑠𝑖𝑛 𝑥
(viii)
𝑥+𝑠𝑖𝑛 𝑥
𝑥+𝑐𝑜𝑠 𝑥
(ix)
𝑠𝑖𝑛 𝑥+𝑐𝑜𝑠𝑥
𝑠𝑖𝑛 𝑥−𝑐𝑜𝑠 𝑥
Findthe derivativesoffollowing:
sin(x2
+5) log(sinx) 𝑒√ 𝑥 log(secx) cos-1
(2x+5) cos(logx) tan2
x
sin3
x √𝑎2 + 𝑥2 (x2
+3x+1)10
tan(2x) sin-1
(5x) sin(sin-1
x) logsinx2
1
√𝑎2−𝑥2
log(secx+tanx) sin-1
(x3
) e xtanx
sec(logxn
) log(x+√𝑎2 + 𝑥2)

4. eax
cos(bx+c) tan-1
(ex
) sin2
x cos3
x (tan-1
x)2
e3x
cos5x √
𝑎+𝑥
𝑎−𝑥
log(cosecx-cotx) logtan(
𝜋
4
+
𝑥
2
) √𝑙𝑜𝑔 {𝑠𝑖𝑛(
𝑥
3
2
− 1)}