S0NAWANE ACADEMY
MATHS COACHING
CHAPTER- limits
JOB SHEET NO- -01
DEFINITION OF LIMIT:
lim 𝑓(𝑥) exists iff
𝑥→𝑎
lim
lim
𝑓(𝑥) =
𝑓(𝑎 − ℎ) exists,
𝑥 → 𝑎−
ℎ→0
lim
lim
(2) R.H.L.=Right hand limit= 𝑓(𝑎+ ) =
𝑓(𝑥) =
𝑓(𝑎 + ℎ) exists,
𝑥 → 𝑎+
ℎ→0
lim
lim
(3)
𝑓(𝑥).
− 𝑓(𝑥) =
𝑥→𝑎
𝑥 → 𝑎+
lim
lim
If lim 𝑓(𝑥) exists then lim 𝑓(𝑥) =
𝑓(𝑥) =
𝑓(𝑥).
𝑥→𝑎
𝑥→𝑎
𝑥 → 𝑎−
𝑥 → 𝑎+
(1) L.H.L.=Left hand limit = 𝑓(𝑎− ) =
Q.1
1] Evaluate: lim 𝑓(𝑥) where 𝑓(𝑥) =
𝑥→𝑎
𝑥 2 −4
at 𝑥 = 2
𝑥−2
|𝑥−4|
, 𝑥≠4
2] Evaluate: lim 𝑓(𝑥) where 𝑓(𝑥) = { 𝑥−4
𝑎𝑡 𝑥 = 4
𝑥→𝑎
0,
𝑥=4
1 + 𝑥2, 0 ≤ 𝑥 ≤ 1
3]Evaluate: lim 𝑓(𝑥) where 𝑓(𝑥) = {
𝑎𝑡 𝑥 = 1
𝑥→𝑎
2 − 𝑥,
𝑥>1
𝑥−|𝑥|
, 𝑥≠0
4]If 𝑓(𝑥) = { 𝑥
show that lim 𝑓(𝑥) does not exist.
𝑥→0
2,
𝑥=0
5𝑥 − 4, 0 < 𝑥 ≤ 1
5]If 𝑓(𝑥) = { 3
show that lim 𝑓(𝑥) exists.
𝑥→1
4𝑥 − 3𝑥, 1 < 𝑥 < 2
6] Discuss the existence of each of the following limits.
1
1
(i)
lim 𝑥
(ii) lim |𝑥|
𝑥→0
𝑥→0
𝑐𝑜𝑠𝑥,
𝑥>0
7] Let 𝑓(𝑥) = {
find the value of constant k ,
𝑥 + 𝑘,
𝑥<0
given that lim 𝑓(𝑥) exists.
𝑥→0
8]Let 𝑓(𝑥) be function defined by
4𝑥 − 5, 𝑥 ≤ 2
𝑓(𝑥) = {
find 𝛼 if lim 𝑓(𝑥) exists.
𝑥 − 𝛼,
𝑥>2
𝑥→2
𝑚𝑥 2 + 𝑛, 𝑥 < 0
9] If 𝑓(𝑥) = {𝑛𝑥 + 𝑚,
0≤𝑥≤1
3
𝑛𝑥 + 𝑚, 𝑥 > 1
For what values of integers m and n does the
limits lim 𝑓(𝑥) and lim 𝑓(𝑥) exists.
𝑥→0
𝑥→1
[NCERT]
LIMITS
|𝑥| + 1, 𝑥 < 0
𝑥 = 0 for what value of a does lim 𝑓(𝑥)exist ? [NCERT]
10] If 𝑓(𝑥) = { 0,
𝑥→0
|𝑥| − 1, 𝑥 > 0
𝑎 + 𝑏𝑥, 𝑥 < 1
11] Suppose 𝑓(𝑥) = { 4,
𝑥=1
𝑏 − 𝑎𝑥, 𝑥 > 1
And if lim 𝑓(𝑥) = 𝑓(1) . what are possible values of a and b?
[NCERT]
𝑥→1
JEE MAIN LEVEL EXAMPLES
(1) Find left hand and right hand limits of the greatest integer function
𝑓(𝑥) = [𝑥]= greatest intger less than or equal to , at 𝑥 = 𝑘 ,
where 𝑘 is an ineteger. Also show that lim 𝑓(𝑥) does not exists.
𝑥→𝑘
2] Prove that lim+[𝑥] = [𝑎] for all ∈ 𝑅 , where [. ] denotes the greatest integer function.
𝑥→𝑎
𝑒 1/𝑥 −1
3] Show that lim 𝑒 1/𝑥 +1 does not exist.
𝑥→0
4] If 𝑓is an odd function and if lim 𝑓(𝑥) exists. Prove that this limit must be zero.
𝑥→0
5] If 𝑓is an even function then prove that lim− 𝑓(𝑥) = lim+ 𝑓(𝑥) exists.
𝑥→0
𝑥→0
HOME WORK
CET / BOARD LEVEL
𝑥
1. Show that lim |𝑥| does not exist.
[NCERT]
𝑥→0
2𝑥 + 3, 𝑥 ≤ 2
2. Find k so that lim 𝑓(𝑥) may exist, where (𝑥) = {
.
𝑥 + 𝑘, 𝑥 > 2
𝑥→2
1
3. Show that lim 𝑥 does not exist.
𝑥→0
4. Let 𝑓(𝑥) be a function defined by 𝑓(𝑥) =
3𝑥
{|𝑥||2𝑥
, 𝑥≠0
0,
𝑥=0
>0
. Prove that lim 𝑓(𝑥) does not exist.
<0
𝑥→0
>0
prove that lim 𝑓(𝑥) does not exist.
<0
𝑥→2
4,
if 𝑥 > 1
7. Find lim 𝑓(𝑥) where 𝑓(𝑥) = {
𝑥 + 1, if 𝑥 < 1
𝑥→1
2𝑥 + 3, 𝑥 ≤ 0
8. If 𝑓(𝑥) = {
find lim 𝑓(𝑥) and lim 𝑓(𝑥)
[NCERT]
3(𝑥 + 1), 𝑥 > 0
𝑥→0
𝑥→1
𝑥 2 − 1, 𝑥 ≤ 1
9. Find lim 𝑓(𝑥), if (𝑥) = { 2
.
[NCERT]
𝑥→1
−𝑥 − 1, 𝑥 > 1
|𝑥|
, 𝑥≠0
10. Evaluate lim 𝑓(𝑥), where 𝑓(𝑥) = { 𝑥
[NCERT]
𝑥→0
0, 𝑥 = 0
11. Let 𝑎1 , 𝑎2 , … … . , 𝑎𝑛 be fixed real number such that 𝑓(𝑥) = (𝑥 − 𝑎1 )(𝑥 − 𝑎2 ) … … . (𝑥 − 𝑎𝑛 )
what is lim 𝑓(𝑥)? for 𝑎 ≠ 𝑎1 , 𝑎2 , … … 𝑎𝑛 compute lim 𝑓(𝑥).
[NCERT]
𝑥 + 1,
5. Let𝑓(𝑥) = {
𝑥 − 1,
𝑥 + 5,
6. Let 𝑓(𝑥) = {
𝑥 − 4,
if 𝑥
if 𝑥
if 𝑥
if 𝑥
𝑥→𝑎1
1
𝑥→0
12. Find lim+ 𝑥−1
𝑥→1
13. Evaluate the following one sided limits.
𝑥−3
𝑥−3
i) lim+ 𝑥 2 −4
ii) lim− 𝑥 2 −4
𝑥→2
2
𝑥→2
(iii) lim+
𝑥→0
1
3𝑥
(iv) lim +
𝑥→−8
BY PROF. SUBHASH SONAWANE| SONAWANE ACADEMY-7722003388
2𝑥
𝑥+8
LIMITS
2
v) lim+
1
𝑥5
𝑥→0
ix) lim +
𝑥→−2
vi) lim+ tan 𝑥
vii) lim + sec 𝑥
x) lim−(2 − cot 𝑥)
(xi) lim−1 + 𝑐𝑜𝑠𝑒𝑐 𝑥
𝜋
2
𝑥→
𝑥 2 −1
2𝑥+4
𝑥→−
𝑥→0
viii) lim−
𝑥→0
𝜋
2
𝑥 2 −3𝑥+2
𝑥 3 −2𝑥 2
𝑥→0
JEE MAIN LEVEL
1
14. Show that lim 𝑒 −𝑥 does not exist
15. Find
𝑥→0
(i) lim [𝑥]
(iii) lim [𝑥]
(ii) lim5 [𝑥]
𝑥→2
𝑥→1
𝑥→
2
16. Prove that lim+[𝑥] = [𝑎] for all 𝑎 ∈ 𝑅. Also prove that lim− [𝑥] = 0
𝑥→𝑎
17. Show that lim−
18. Find
𝑥
[𝑥]
𝑥→2
𝑥
lim
it is
𝑥→3+ [𝑥]
𝑥→−
𝑥→2
equal to lim−
19. Find lim5[𝑥]
𝑥→1
𝑥
≠ lim+ [𝑥]
𝑥→3
𝑥
[𝑥]
2
𝑥 − [𝑥], 𝑥 < 2
20. Evaluate lim 𝑓(𝑥) (if it exists), where 𝑓(𝑥) = { 4,
𝑥=2
𝑥→2
3𝑥 − 5, 𝑥 > 2
1
21. Show that lim sin 𝑥 does not exist.
𝑥→0
𝑘 cos 𝑥
,
𝜋−2𝑥
22. Let 𝑓(𝑥) = {
3,
where 𝑥 ≠
where 𝑥 ≠
𝜋
2
𝜋
2
𝜋
2
and if lim𝜋 𝑓(𝑥) = 𝑓 ( ), find the value of k.
𝑥→
2
Answer:
2) 𝑘 = 5
7) 4
8) 3, does not exist
9) does not exist
11) 0, (𝑎 − 𝑎1 )(𝑎 − 𝑎2 ) … … . . (𝑎 − 𝑎1 )
2) ∞
13)
i)−∞ (ii) ∞
iv)−∞ (v) ∞
vi) ∞ vii)−∞ viii)−∞
ix) ∞ x) ∞ xi)−∞
15)
i) does not exist
(ii)2
(iii)does not exist
18)1, No
19) −3
20) 1
22) 𝑘 = 6
3
BY PROF. SUBHASH SONAWANE| SONAWANE ACADEMY-7722003388
10) does not exist
(iii) ∞