Limits of Function (1)

LIMITS # 1

DEFINITIONS 1.

AND

R E S U LT S

Limit of a function f(x) is said to exist as , x  a when , Left hand limit = Right hand limit Limit f (a  h) = Limit f (a + h) = some finite value M . h0 h0

Note that we are not interested in knowing about what happens at x = a . Also note that if L.H.L. & R.H.L. are both tending towards '  ' then it is said to be infinite limit . Remember , Limit  x  a x a 2.

Fundamental Theorems On Limits : Let Limit x  a f (x) = l &

Limit g (x) = m . If l & m exists then : x a

(i)

Limit f (x) ± g (x) = l ± m x a

(iii)

Limit f ( x)

(iv)

Limit k f(x) = k Limit f(x) ; where k is a constant . x a x a

(v)

Limit f [g(x)] = f  Limit g (x) = f (m) ; provided f is continuous at g (x) = m . x a  xa 

x a

g ( x)

=

Limit f(x) . g(x) = l . m x a

(ii)

l , provided m  0 m

For example let us assume that g (x) = sgn (x) & f (x) = c . Then Limit x  0 f [ g (x) ] = c even if ' g ' is discontinuous at x = 0 . 3. (a)

Standard Limits : 1 1 Limit sinx = 1 = Limit tan x = Limit tan x = Limit sin x x 0 x 0 x 0 x 0

x

x

x

x

[ Where x is measured in radians ] (b)

Limit (1 + x) 1/x = e = Limit  1  1  x 0 x  x

x n note however that Limit n   (1  h) = 0 &

Limit (1 + h)n , where h > 0 . n  Limit f(x) = 1 & x a

Limit (x) = , then ; x a

Limit  f ( x ) ( x )  e Limit xa x a

(c)

If

(d)

Limit (x) = B (a finite quantity) then ; If Limit x  a f(x) = A > 0 & x a

 ( x )[ f ( x ) 1]

Limit [f(x)] (x) = ez where z = Limit (x) . ln[f(x)] = e BlnA = AB x a x a

(e)

x x Limit a  1 = 1n a (a > 0). In particular Limit e  1 = 1 x 0 x 0

(f)

n n Limit x  a  n a n 1 . x a

x

x

xa

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LIMITS # 2

General Note : All the above standard limits can be generalized if ' x ' is replaced by f (x) . sin f ( x) e.g. Limit = 1 provided Limit xa x  a f (x) = 0 & so & so on . f ( x)

4.

Sandwich Theorem or Squeeze Play Theorem : Limit h(x) If f(x)  g(x)  h(x)  x & Limit x  a f(x) = l = x a

5.

then Limit x  a g(x) = l .

Indeterminant Forms : 0  , , 0 x  , 0 ,  ,    and 1 . 0 

Note : (i) Note here that ' 0 ' doesn't means exact zero but represent a value approaching towards zero similary fo ' 1 ' and infinity . (ii) += (iii)  x  =  (iv) (a/) = 0 if a is finite (v)

a 0

(vi)

a b = 0 , if & only if a = 0 or b = 0 and a & b are finite .

6.

(a)

To evaluate a limit, we must always put the value where ' x ' is approaching to in the function . If we get a deterninate form, than that value becomes the limit otherwise if an indeterminant form comes . This apply one of the following methods : Factorisation (b) Rationalisation or double rationalisation

(c)

Substitution

(e)

Expansions of functions . The following expansions must be remembered :

(i)

a x 1 

(iii)

ln (1+x) = x 

(v)

cos x  1 

(vii)

tan -1x = x 

x3 x5 x7   ....... 3 5 7

(ix)

sec-1x = 1 

x 2 5x 4 61x 6   ...... 2! 4! 6!

is not defined for any a  R .

(d)

x 1n a x 2 1n 2 a x 3 1n 3a   ......... a  0 1! 2! 3! x2 x3 x 4   ......... for  1  x  1 2 3 4

x2 x4 x6   ...... 2! 4! 6!

EXERCISE 1.

  Limit  x  2  x  x  2

(A) e4 2.

(viii)

x x2 x3   ............ 1! 2! 3!

(ii)

ex  1 

(iv)

sin x  x 

(vi)

tan x = x 

sin -1x = x 

x3 x5 x7   ....... 3! 5! 7! x 3 2x 5  ........ 3 15

12 3 12 .32 5 12 .32 .52 7 x  x  x ....... 3! 5! 7!

I

x 1

= (B) e 4

(C) e2

(D) none

Limit (1  x + [x  1] + [1  x]) = where [x] denotes greatest integer function . x1

(A) 0 3.

Using standard limits

(B) 1

(C)  1

(D) does not exist

If ,  are the roots of the quadratic equation ax 2 + bx + c = 0 then

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LIMITS # 3



x

(x   )

(A) 0 4.

=

1  cos ax 2  bx  c

Limit

2

(B)

6.

(B) 3

Limit



(A)

ab

x

a2   2 (  ) 2

(C) 1

(D) zero

(C) ab

(D) none

(B) 1/2

(C)  1

(D) none

(B)  2 cos 2

(C) 2 sin 2

(D)  2 sin 2

(B)  1/3

(C) 0

(D) 3

(B)

5 x  2

Limit x 1

2x  1

ab 2

=

Limit cos 2  cos 2x = x  1 x2  x

x2  2 = 3x  6

Limit x  

(A) 1/3 9.

(D) 



(A) 2 cos 2 8.

a2   2 (  ) 2

(x  a ) (x  b)  x =

(A) 2 7.

(C)

n 1 n 2n Limit 5  3  2 = n 5 n  2 n  32 n  3

(A) 5 5.

1   2 (  ) 2

 sin[x ] if [x]  0 where [x] denotes the greatest integer less than or equal  0 if [x]  0

 If f(x) =  [x]

to x, then Limit x  0 f(x) equals : (A) 1 10.

Limit

x

x0

sin x

(D) none

(B) 1

(C)  1

(D) none of these

(C) 0

(D) none

(B) e2

(C) e

(D) none

(B) 1/6

(C) 1/12

(D) 1/8

1 The value of Limit x0

1  cos 2x is : 2

x

(A) 1 12.

(C)  1

=

(A) 0 11.

(B) 0

Limit

x  0

(B)  1

1  tan x  2

5/ x

(A) e5

=

2

13.

Limit x0

e  x / 2  cos x = x 3 . sin x

(A) 1/4 14.

Limit  cos mx x0 2

(A) e  m n / 4

n / x2

= 2

(B) e  m n / 2

(C) e  mn

2

/2

(D) e  mn

2

/4

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LIMITS # 4

15.

1 13 x  2 3 x  ......  n 3 x equals If [x] denotes the greatest integer  x, then Limit 4 n n

(A) x/2 16.

x 3 . sin x1  x  1

Limit

x2  x  1

x

(A) 0 18.

If Limit x0

Limit  tan x  x0  x   

(C) 1

(D) none

(C) 1

(D) none

=

(B) 2

(C) 1

(B) e2/3

(C) e1/3

x0

(D) none

(B) 1/2

(C) 1/3

(D) none

(B)

(C) 1

(D) 0

(C) 9

(D)  9

1 Limit cos (1  x) =  x0 x

1

Limit x3

2

2

(x 3  27) log (x  2) (x 2  9)

=

(B) 8

n Limit x = 0 (n integer) for : x ex

(A) no value of n (C) only negative values of n 24.

(D)  2

is :

(A)  8 23.

(B) 0

2 (tan x  sin x)  x3 = x5

Limit

(A)

22.

(D) x/4

1/ x 2

(A) 1/4 21.

(C) x/6

sin 2x  a sin x is finite, then the value of a is : x3

(A) e 20.

(B) x/3

(B) 1/2

(A)  1 19.

 

tan x  ex Limit e = x0 tan x  x

(A) 1/2 17.

   

(B) all values of n (D) only positive values of n

Limit sin 1 (sec x) . x0

(A) is equal to /2

(B) is equal to 1

(C) is equal to zero

(D) none of these

2

25.

Limit x5

x  9x  20 where [x] is the greatest integer not greater than x : x  [x]

(A) is equal to 1 26.

(B) 0

(D) none

2 2/x 3 If Limit x  0 (1 + a x + b x ) = e , then :

(A) a = 3 , b = 0 1

27.

(C) 4

(B) a = 3/2 , b = 1

(C) a = 3/2 , b = 4

(D) a = 2 , b = 3

1

Limit sin x  tan x = x0 x3

(A) 0

(B) 1/2

(C) 1

(D) none

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LIMITS # 5 x

28.

2 Limit  x  2x  1  = x 2  x  4x  2 

(A) 1 29.

31.

(x  a )

3 (a + 2)2/3 5

(B)

(B) 1/11

Limit ( n  2) !  ( n  1) ! , n  ( n  3) !

Limit n

sin 2 x  6 sin x  2  is equal to : 

(C) 1/12

(D) 1/8

(C) 1/4

(D) 1/2

(C) 2

(D)  1

       sin   has the value equal to :  4 n  4 n

n cos 

Limit x   x   sin x

(B) /4

(C) /6

(D) none

(B) is equal to 1

(C) is equal to 

(D) does not exist

(B) 0

(C) 1

(D) none of these

(C) e 2

(D) e1

(C) 12 p (log 4)3

(D) 27 p (log 4)2

:

x x Limit 3  5 = x 7x  11x

Limit  cot x0  

    x  4 

cos ec x

(A) e 1 38.

(D) none of these

nN=

(B) 1

(A)  1 37.

3 (a + 2)3/2 5

x

(B) 1/6

(A) is equal to  1 36.

(D)

cos (sin x)  cos x The value of Limit is equal to : x0 4

(A) /3 35.

5 (a + 2)2/3 3



(A) 0 34.

(C)

2 2  The value of Limit x   / 2 tan x  2 sin x  3 sin x  4 

(A) 1/5 33.

5 (a + 2)3/2 3

The limiting value of (cos x) 1/sin x as x  0 is : (A) 1 (B) e (C) 0

(A) 1/10 32.

(D) e

(x  2)5/ 3  (a  2)5/ 3 If Limit = xa

(A)

30.

(C) e2

(B) 2

Limit x0

= (B) e2

(4x  1)3 2 sin  xp  log 1  x  3 

(A) 9 p (log 4)

=

(B) 3 p (log 4)3

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LIMITS # 6

39.

 ay by    exp  x  n 1  x    exp  x  n 1  x          Limit  Limit  = y0 y x   

(A) a + b 40.

(B) a  b

Limit

1  sin 2 x

x  /4

  4x

(A) 2 41.

(D) does not exist

(B) 

(C) 1

(D) none of these

(C) 4550

(D)  5050

(C) 2

(D)  2

(C) 2

(D) 1

(B) 5050

Limit log (2  x)  log 0.5 = x0 x

(B)  1/2

Limit x2



=

sin e x  2  1 log (x  1)

(B)  1

(A) 0 45.

(C) 0

 100    x k   100 Limit  k  1  = x 1 x 1

(A) 1/2 44.

(B)  2

=

x  cos 2 x

(A) 0 43.

(D)  (a + b)

x  sin x

Limit x

(A) 0

42.

(C) b  a

2 Limit  x0  



 3





3 sin   x  cos   x 6 6 x

(A)  1/3

EXERCISE

3 cos x  sin x

(B) 2/3



  

 = 

(D)  4/3

(C) 4/3

II

Level I Evaluate the following limits : x 3  7 x 2  15 x  9

1.

Limit

3.

Limit (x  y ) sec (x  y )  x sec x y0 y

4.

x

5.

x x x Limit 10  2  5  1 x0 x tan x

6.

x

x3

x 4  5 x 3  27 x  27

2.

Limit xa

a  2x 

3a  x  2 x

Limit x  tan  1   Limit

3x

 x  1       x  4 4

  x x x  

 x 

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LIMITS # 7

7.

2  cosx  1 x  /2 x x  2

9.

Limit

10.

 x2  1    a x  b = 2 , find values of a & b . If xLimit   x 1

Limit

x0



Limit

8.



x

 cos 



x  1  cos

 x

1  x sin x  cos 2 x tan 2 x 2





Level II Evaluate the following limits : 31

x2  4 1  2 x

11.

Limit

13.

Limit

15.

 x2  Limit  e  cos x  x0   x2  

16.

Limit x 3  x 2  1  x 4  x 2     

18.

17.

x0

x0

x  x2 31

tan  1 3 x  3 1  sin  1 3 x

1  sin  1 2 x 

1/ x Limit e  (1  x ) x0 tan x

12.

x 3 x 6 Limit 2  2

14.

1  tan  1 2 x

x

x2

x x x Limit 27  9  3  1 x0 5  4  cos x

Limit

(1n(1  x)  1n2)(3.4 x 1  3x )

x1

[(7  x) 3  (1  3x ) 2 ].sin(x  1)

19.

Limit n  n2  ( n  1)  n  1   n  1  .......  n  1          n    2  22  2n  1   

20.

Limit

21.

Limit cos x cos x cos x ........ cos x n  2 4 8 2n

22.

2  x  21  x

1

1

n

 x2 x2 x2 x2  1  cos  cos  cos cos   x0 2 4 2 4  x 8 

Limit

8

  2 x 5 tan  1 2   3 x  7  x 

x  

x

3

23.

7 x 8



Limit

log sec(x / 2) cos x

x0

logsec x cos (x / 2)



cos 2 1  cos 2 1  cos2 ....... cos2 (x )  .......    x0   x  4  2  sin     x   

24.

Limit

25.

Limit

[ x ]  [ 22 . x ]  [32 . x ] 

n 

n3

.......  [ n 2 . x ]

, where [ ] denotes greatest integer

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LIMITS # 8

26.

(x  h ) x  h  x x Show that Limit = x x . (1 + l n x) . h 0

27.

2 x 2 x 2 x   Limit  Limit  [1 (sin x) ]  [2 (sin x) ]  .......  [n (sin x) ]   Evaluate x  0  n     , n3    

h

where [ . ] denotes the greatest integer function . x n f (x )  g(x ) Limit 28. n xR n x 1

29.

Limit log (x).log (x  1).log (x  2).log (x  3).....log (x 5 ) ; where k = x 5  1 . x 1 x x 1 x2 k

30.

Limit 1 tan x + 1 tan x + 1 tan x + ...... + 1 tan x . n 2 2 22 22 2n 23 23 2n

31.

Find the values of a & b so that :

(i)

Limit

(ii)

x 

(1  a x sin x)  (b cos x)

x0

x4

may find to a definite limit .

Limit  x 2  x  1  a x  b = 0 .  

x  

Evaluate the following limits : sin x

32.

34.

Limit  sin x  x  sin x  x  0  x  Limit x  7/ 2

2 x

2



 9x  8

36.

Limit  x  1  cos x   x  0   x

38.

 2  Limit  3 x  4    2 x   3x  6 

40.

1/ x   Limit  1  x   x0   e  

33. cot (2 x  7)

1 2 log (1  x)

Limit

1  x 

Limit

x m (log x)n

x  1

37.

x  0

39.

 x  2

m, n N

9 x2  4

Limit

(sec x)cot x

1/ x

41.

42.

Limit (1 + tan 2 x)cot 3 x x0

44.

Limit  tan    log x  log x   x 1  4  

46.

x Limit  x  a 2    a

x 2a

Limit  log x  x 

1/ x

   x    sec 2     2  bx

43.

Limit  sin 2       x0   2  a x  

45.

Limit  tan  1 (x) . 4  x2  1 x 1    

1

tan

 2  x

35.

1/ x

2x

Limit  x  x 

1

47.

2/ x x x  x Limit  a  b  c  , where a , b , c > 0  x0  3  

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LIMITS # 9

48.

1/ x 1/ x 1/ x   1/ x Limit  a 1  a 2  a 3  .......  a n  x 

 

nx

, where a1 , a2 , a3 , ....... , an > 0 .

 

n

From Q. Nos. 49  61 , evaluate the following limits without using L` Hospital rule and series expansion . 49.

Limit tan x  sin x x0 x3

50.

2 2 Limit tan x  x x0 x 2 tan 2 x

51.

(1  x )  1 Limit   n 1  x    x0  x x2  

52.

1 1 Limit tan x  sin x x0 sin 3 x

53.

Limit x  sin x x0 x3

54.

x Limit e  1  x x0 x2

55.

x x cos x Limit e  e x0 x  sin x

56.

57.

Limit

4 2   cos x  sin x 

x  /4

1  sin 2 x

x   1

 

cos  1 x x1

5

58.

x x Limit (cos  )  (sin )  cos 2  , x 4 x4

59.

x x Limit e  e  2 x x0 x  sin x

61.

Limit

 

 2

a   0 ,  60.

Limit x  x 2 l n x 

1  1    x

x   n  1  x 2  x   Limit x0 3 x

62.

Through a point A on a circle, a chord AP is drawn & on the tangent at A a point T is taken such that AT = AP . If TP produced meet the diameter through A at Q, prove that the limiting value of AQ when P moves upto A is double the diameter of the circle .

63.

If s n be the sum of n terms of the series, sin x + sin 2x + sin 3x + ..... + sin nx then show s1  s 2  ......  s n 1 x that Limit = cot (x  2 k , k  I) n n

64.

Let f(x) =

2

sin 1 (1 {x}).cos 1 (1  {x}) 2{x} . (1 {x})

2

Limit then find Limit x  0  f(x) and x  0  f(x) ,

where {.} denotes the fractional part function . 65.

  2m (n !  x  where x  R . Prove that Let f (x) = mLimit    Limit cos n





 1 if x is rational .  0 if x is irrational

f (x) = 

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LIMITS # 10

23  1

2 . Prove that Limit . n   Pn =

A sequence of numbers x n is determined by the equality x n 

3 1

3

4 1

. ........ .

n3  1

67.

2 1

3

.

43  1

Let Pn =

3

.

33  1

66.

3

n 1

3

x n 1  x n  2 & the values 2

x  2 x1 x 0 and x 1 Prove that nLimit x = 0 .  n 3

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LIMITS # 11

ANSWER EXERCISE

SHEET

I

1. A

2. C

3. C

4. D

5. B

6. B

7. C

8. B

9. D

10. B

11. D

12. A

13. C

14. B

15. D

16. C

17. C

18. D

19. C

20. A

21. B

22. C

23. B

24. D

25. D

26. BC

27. B

28. C

29. C

30. A

31. C

32. B

33. A

34. B

35. D

36. B

37. C

38. B

39. B

40. D

41. C

42. B

43. A

44. D

45. C

EXERCISE 1.

2 9

2.

6.

1 2

7.

12.

e 2

18.  25.

II

2 3 3

2 n 2 

13.  1

9 4 1n 4 e

x 3

19. e2 27.

3. x tan x sec x + sec x 8. zero

9. 6

14. 8 20.

1 32

33. e 4

34. e5/2

10. a = 1 , b =  3

3 2

16. 8 5 (log 3)2

21.

sin x x

22. 

1 

1 2

11. 17.

23. 16

1 4 2

24.

2

1 3

1  cot x x

30.

5. log5 log 2

15.

28. f (x) when x> 1 ; g(x) when x < 1 ; 29. 5

3 2

4. 

31. (i) a =  35. e1

g(x)  f (x) when x = 1 & not defined when x = 1 2

1 , b=1 2

36. e 1/2

(ii) a = 1 , b =

1 2

32. e 1

38. e 6

37. 0

39. 1

2  a

40. e

1/2

46. e2/ 52. 

1 2

41. 1

42. e

2/3

43. e

b2

47. (a b c)2/3 48. (a1 a2 a3 ....... an) 53.

1 6

54.

1 2

58. cos 4  l n (cos )  sin 4  l n (sin )

55. 0 59. 2

44. e2 49. 56. 60.

1 2

45. e1/ 50.

1 2

1 2

2 3

51.

1 2

57. 5 2 61.

1 6

64.  , 

2 2 2

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LIMITS # 12

EXERCISE

III

A 1 1. A= h 2 rh  h 2 , Limit 3  h0 P 128r

2. e5

3. D

6. D

10. C

11

7. C

8. C

9. B

4. e² C

12

5. e² 1

2 

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