1. The document provides examples of evaluating limits of functions as the variable approaches certain values. It includes limits of piecewise defined functions, trigonometric functions, rational functions, and composite functions. 2. Methods demonstrated include evaluating left and right hand limits, simplifying expressions using trigonometric identities, factorizing polynomials, and applying definitions of limits. 3. The document contains over 25 examples of limit evaluations and derivative calculations using basic limit laws and derivative rules.

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)}