Maharashtra State BoardHSC Science (General) 11th
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Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board chapter 7 - Limits [Latest edition]

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Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board - Shaalaa.com
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Chapter 7: Limits

Exercise 7.1Exercise 7.2Exercise 7.3Exercise 7.4Exercise 7.5Exercise 7.6Exercise 7.7Miscellaneous Exercise 7
Exercise 7.1 [Pages 138 - 139]

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board Chapter 7 Limits Exercise 7.1 [Pages 138 - 139]

Exercise 7.1 | Q I. (1) | Page 138

Evaluate the following limit:

`lim_(z -> -3) [sqrt("z" + 6)/"z"]`

Exercise 7.1 | Q I. (2) | Page 138

Evaluate the following limit:

`lim_(y -> -3) [(y^5 + 243)/(y^3 + 27)]`

Exercise 7.1 | Q I. (3) | Page 138

Evaluate the following limit:

`lim_(z -> -5)[((1/z + 1/5))/(z + 5)]`

Exercise 7.1 | Q II. (1) | Page 138

Evaluate the following limit:

`lim_(x -> 3)[sqrt(2x + 6)/x]`

Exercise 7.1 | Q II. (2) | Page 139

Evaluate the following limit:

`lim_(x -> 2)[(x^(-3) - 2^(-3))/(x - 2)]`

Exercise 7.1 | Q II. (3) | Page 139

Evaluate the following limit:

`lim_(x -> 5)[(x^3 - 125)/(x^5 - 3125)]`

Exercise 7.1 | Q II. (4) | Page 139

Evaluate the following limit:

If `lim_(x -> 1)[(x^4 - 1)/(x - 1)]` = `lim_(x -> "a")[(x^3 - "a"^3)/(x - "a")]`, find all possible values of a

Exercise 7.1 | Q III. (1) | Page 139

Evaluate the following limit :

`lim_(x -> 1)[(x + x^2 + x^3 + ......... + x^"n" - "n")/(x - 1)]`

Exercise 7.1 | Q III. (2) | Page 139

Evaluate the following limit :

`lim_(x -> 7)[((root(3)(x) - root(3)(7))(root(3)(x) + root(3)(7)))/(x - 7)]`

Exercise 7.1 | Q III. (3) | Page 139

Evaluate the following limit : 

If `lim_(x -> 5) [(x^"k" - 5^"k")/(x - 5)]` = 500, find all possible values of k.

Exercise 7.1 | Q III. (4) | Page 139

Evaluate the following limit :

`lim_(x -> 0)[((1 - x)^8 - 1)/((1 - x)^2 - 1)]`

Exercise 7.1 | Q III. (5) | Page 139

Evaluate the following limit :

`lim_(x -> 0)[(root(3)(1 + x) - sqrt(1 + x))/x]`

Exercise 7.1 | Q III. (6) | Page 139

Evaluate the following limit :

`lim_(y -> 1)[(2y - 2)/(root(3)(7 + y) - 2)]`

Exercise 7.1 | Q III. (7) | Page 139

Evaluate the following limit :

`lim_(z -> "a")[((z + 2)^(3/2) - ("a" + 2)^(3/2))/(z - "a")]`

Exercise 7.1 | Q III. (8) | Page 139

Evaluate the following limit :

`lim_(x -> 7) [(x^3 - 343)/(sqrt(x) - sqrt(7))]`

Exercise 7.1 | Q III. (9) | Page 139

Evaluate the following limit :

`lim_(x -> 1) [(x + x^3 + x^5 + ... + x^(2"n" - 1) - "n")/(x - 1)]`

Exercise 7.1 | Q IV. (1) | Page 139

In the following example, given ∈ > 0, find a δ > 0 such that whenever, |x – a| < δ, we must have |f(x) – l| < ∈.

`lim_(x -> 2)(2x + 3)` = 7

Exercise 7.1 | Q IV. (2) | Page 139

In the following example, given ∈ > 0, find a δ > 0 such that whenever, |x – a| < δ, we must have |f(x) – l| < ∈.

`lim_(x -> -3) (3x + 2)` = – 7

Exercise 7.1 | Q IV. (3) | Page 139

In the following example, given ∈ > 0, find a δ > 0 such that whenever, |x – a| < δ, we must have |f(x) – l| < ∈.

`lim_(x -> 2) (x^2 - 1)` = 3

Exercise 7.1 | Q IV. (4) | Page 139

In the following example, given ∈ > 0, find a δ > 0 such that whenever, |x – a| < δ, we must have |f(x) – l| < ∈.

`lim_(x -> 1) (x^2 + x + 1)` = 3

Exercise 7.2 [Page 141]

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board Chapter 7 Limits Exercise 7.2 [Page 141]

Exercise 7.2 | Q I. (1) | Page 141

Evaluate the following limit :

`lim_(z -> 2) [(z^2 - 5z + 6)/(z^2 - 4)]`

Exercise 7.2 | Q I. (2) | Page 141

Evaluate the following limit :

`lim_(x -> -3)[(x + 3)/(x^2 + 4x + 3)]`

Exercise 7.2 | Q I. (3) | Page 141

Evaluate the following limit :

`lim_(y -> 0)[(5y^3 + 8y^2)/(3y^4 - 16y^2)]`

Exercise 7.2 | Q I. (4) | Page 141

Evaluate the following limit :

`lim_(x -> -2) [(-2x - 4)/(x^3 + 2x^2)]`

Exercise 7.2 | Q I. (5) | Page 141

Evaluate the following limit :

`lim_(x -> 3) [(x^2 + 2x - 15)/(x^2 - 5x + 6)]`

Exercise 7.2 | Q II. (1) | Page 141

Evaluate the following limit :

`lim_(u -> 1) [(u^4 - 1)/(u^3 - 1)]`

Exercise 7.2 | Q II. (2) | Page 141

Evaluate the following limit :

`lim_(x -> 3) [1/(x - 3) - (9x)/(x^3 - 27)]`

Exercise 7.2 | Q II. (3) | Page 141

Evaluate the following limit :

`lim_(x -> 2)[(x^3 - 4x^2 + 4x)/(x^2 - 1)]`

Exercise 7.2 | Q II. (4) | Page 141

Evaluate the following limit :

`lim_(Deltax -> 0) [((x + Deltax)^2 - 2(x + Deltax) + 1 - (x^2 - 2x + 1))/(Deltax)]`

Exercise 7.2 | Q II. (5) | Page 141

Evaluate the following limit :

`lim_(x -> sqrt(2)) [(x^2 + xsqrt(2) - 4)/(x^2 - 3xsqrt(2) + 4)]`

Exercise 7.2 | Q II. (6) | Page 141

Evaluate the following limit :

`lim_(x -> 2) [(x^3 - 7x + 6)/(x^3 - 7x^2 + 16x - 12)]`

Exercise 7.2 | Q III. (1) | Page 141

Evaluate the following limit :

`lim_(y -> 1/2) [(1 - 8y^3)/(y - 4y^3)]`

Exercise 7.2 | Q III. (2) | Page 141

Evaluate the following limit :

`lim_(x -> 1) [(x - 2)/(x^2 - x) - 1/(x^3 - 3x^2 + 2x)]`

Exercise 7.2 | Q III. (3) | Page 141

Evaluate the following limit :

`lim_(x -> 1) [(x^4 - 3x^2 + 2)/(x^3 - 5x^2 + 3x + 1)]`

Exercise 7.2 | Q III. (4) | Page 141

Evaluate the following limit :

`lim_(x -> 1) [(x + 2)/(x^2 - 5x + 4) + (x - 4)/(3(x^2 - 3x + 2))]`

Exercise 7.2 | Q III. (5) | Page 141

Evaluate the following limit :

`lim_(x -> "a")[1/(x^2 - 3"a"x + 2"a"^2) + 1/(2x^2 - 3"a"x + "a"^2)]`

Exercise 7.3 [Page 143]

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board Chapter 7 Limits Exercise 7.3 [Page 143]

Exercise 7.3 | Q I. (1) | Page 143

Evaluate the following limit :

`lim_(x -> 0)[(sqrt(6 + x + x^2) - sqrt(6))/x]`

Exercise 7.3 | Q I. (2) | Page 143

Evaluate the following limit :

`lim_(x -> 3)[(sqrt(2x + 3) - sqrt(4x - 3))/(x^2 - 9)]`

Exercise 7.3 | Q I. (3) | Page 143

Evaluate the following limit :

`lim_(y -> 0)[(sqrt(1 - y^2) - sqrt(1 + y^2))/y^2]`

Exercise 7.3 | Q I. (4) | Page 143

Evaluate the following limit :

`lim_(x -> 2) [(sqrt(2 + x) - sqrt(6 - x))/(sqrt(x) - sqrt(2))]`

Exercise 7.3 | Q II. (1) | Page 143

Evaluate the following limit :

`lim_(x -> "a") [(sqrt("a" + 2x) - sqrt(3x))/(sqrt(3"a" + x) - 2sqrt(x))]`

Exercise 7.3 | Q II. (2) | Page 143

Evaluate the following limit :

`lim_(x -> 2) [(x^2 - 4)/(sqrt(x + 2) - sqrt(3x - 2))]`

Exercise 7.3 | Q II. (3) | Page 143

Evaluate the following limit :

`lim_(x -> 2)[(sqrt(1 + sqrt(2 + x)) - sqrt(3))/(x - 2)]`

Exercise 7.3 | Q II. (4) | Page 143

Evaluate the following limit :

`lim_(y -> 0) [(sqrt("a" + y) - sqrt("a"))/(ysqrt("a" + y))]`

Exercise 7.3 | Q II. (5) | Page 143

Evaluate the following limit :

`lim_(x -> 0)[(sqrt(x^2 + 9) - sqrt(2x^2 + 9))/(sqrt(3x^2 + 4) - sqrt(2x^2 + 4))]`

Exercise 7.3 | Q III. (1) | Page 143

Evaluate the following limit :

`lim_(x -> 1) [(x^2 + xsqrt(x) - 2)/(x - 1)]`

Exercise 7.3 | Q III. (2) | Page 143

Evaluate the Following limit :

`lim_(x -> 0) [(sqrt(1 + x^2) - sqrt(1 + x))/(sqrt(1 + x^3) - sqrt(1 - x))]`

Exercise 7.3 | Q III. (3) | Page 143

Evaluate the Following limit :

`lim_(x -> 4) [(x^2 + x - 20)/(sqrt(3x + 4) - 4)]`

Exercise 7.3 | Q III. (4) | Page 143

Evaluate the Following limit :

`lim_(z -> 4) [(3 - sqrt(5 + z))/(1 - sqrt(5 - z))]`

Exercise 7.3 | Q III. (5) | Page 143

Evaluate the Following limit :

`lim_(x -> 0)[3/(xsqrt(9  - x)) - 1/x]`

Exercise 7.4 [Page 148]

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board Chapter 7 Limits Exercise 7.4 [Page 148]

Exercise 7.4 | Q I. (1) | Page 148

Evaluate the following limit :

`lim_(theta -> 0) [(sin("m"theta))/(tan("n"theta))]`

Exercise 7.4 | Q I. (2) | Page 148

Evaluate the following limit :

`lim_(theta -> 0) [(1 - cos2theta)/theta^2]`

Exercise 7.4 | Q I. (3) | Page 148

Evaluate the following limit :

`lim_(x -> 0) [(x*tanx)/(1 - cosx)]`

Exercise 7.4 | Q I. (4) | Page 148

Evaluate the following limit :

`lim_(x ->0)((secx - 1)/x^2)`

Exercise 7.4 | Q II. (1) | Page 148

Evaluate the following limit :

`lim_(x -> 0)[(1 - cos("n"x))/(1 - cos("m"x))]`

Exercise 7.4 | Q II. (2) | Page 148

Evaluate the following limit :

`lim_(x -> pi/6) [(2 - "cosec"x)/(cot^2x - 3)]`

Exercise 7.4 | Q II. (3) | Page 148

Evaluate the following limit :

`lim_(x -> pi/4) [(cosx - sinx)/(cos2x)]`

Exercise 7.4 | Q III. (1) | Page 148

Evaluate the following limit :

`lim_(x -> 0) [(cos("a"x) - cos("b"x))/(cos("c"x) - 1)]`

Exercise 7.4 | Q III. (2) | Page 148

Evaluate the following limit :

`lim_(x -> pi) [(sqrt(1 - cosx) - sqrt(2))/(sin^2 x)]`

Exercise 7.4 | Q III. (3) | Page 148

Evaluate the following limit :

`lim_(x -> pi/4) [(tan^2x - cot^2x)/(secx - "cosec"x)]`

Exercise 7.4 | Q III. (4) | Page 148

Evaluate the following limit :

`lim_(x -> pi/6) [(2sin^2x + sinx - 1)/(2sin^2x - 3sinx + 1)]`

Exercise 7.5 [Pages 150 - 151]

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board Chapter 7 Limits Exercise 7.5 [Pages 150 - 151]

Exercise 7.5 | Q I. (1) | Page 150

Evaluate the following :

`lim_(x -> pi/2) [("cosec"x - 1)/(pi/2 - x)^2]`

Exercise 7.5 | Q I. (2) | Page 150

Evaluate the following :

`lim_(x -> "a") [(sinx - sin"a")/(root(5)(x) - root(5)("a"))]`

Exercise 7.5 | Q I. (3) | Page 150

Evaluate the following :

`lim_(x -> pi) [(sqrt(5 + cosx) - 2)/(pi - x)^2]`

Exercise 7.5 | Q I. (4) | Page 150

Evaluate the following :

`lim_(x -> pi/6) [(cos x - sqrt(3) sinx)/(pi - 6x)]`

Exercise 7.5 | Q I. (5) | Page 150

Evaluate the following :

`lim_(x -> 1) [(1 - x^2)/(sinpix)]`

Exercise 7.5 | Q II. (1) | Page 150

Evaluate the following :

`lim_(x -> pi/6) [(2sinx - 1)/(pi - 6x)]`

Exercise 7.5 | Q II. (2) | Page 150

Evaluate the following :

`lim_(x -> pi/4) [(sqrt(2) - cosx - sinx)/(4x - pi)^2]`

Exercise 7.5 | Q II. (3) | Page 150

Evaluate the following :

`lim_(x -> pi/6) [(2 - sqrt(3)cosx - sinx)/(6x - pi)^2]`

Exercise 7.5 | Q II. (4) | Page 151

Evaluate the following :

`lim_(x -> "a") [(sin(sqrt(x)) - sin(sqrt("a")))/(x - "a")]`

Exercise 7.5 | Q II. (5) | Page 151

Evaluate the following :

`lim_(x -> pi/2) [(cos3x + 3cosx)/(2x - pi)^3]`

Exercise 7.6 [Page 154]

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board Chapter 7 Limits Exercise 7.6 [Page 154]

Exercise 7.6 | Q I. (1) | Page 154

Evaluate the following limit : 

`lim_(x -> 0) [(9^x - 5^x)/(4^x - 1)]`

Exercise 7.6 | Q I. (2) | Page 154

Evaluate the following limit : 

`lim_(x -> 0) [(5^x + 3^x - 2^x - 1)/x]`

Exercise 7.6 | Q I. (3) | Page 154

Evaluate the following limit : 

`lim_(x -> 0)[("a"^x + "b"^x + "c"^x - 3)/sinx]`

Exercise 7.6 | Q I. (4) | Page 154

Evaluate the following limit : 

`lim_(x -> 0) [(6^x + 5^x + 4^x - 3^(x + 1))/sinx]`

Exercise 7.6 | Q I. (5) | Page 154

Evaluate the following limit : 

`lim_(x -> 0) [(8^sinx - 2^tanx)/("e"^(2x) - 1)]`

Exercise 7.6 | Q II. (1) | Page 154

Evaluate the following limit : 

`lim_(x -> 0) [(3^x + 3^-x - 2)/(x*tanx)]`

Exercise 7.6 | Q II. (2) | Page 154

Evaluate the following limit : 

`lim_(x -> 0) [(3 + x)/(3 - x)]^(1/x)`

Exercise 7.6 | Q II. (3) | Page 154

Evaluate the following limit : 

`lim_(x -> 0)[(5x + 3)/(3 - 2x)]^(2/x)`

Exercise 7.6 | Q II. (4) | Page 154

Evaluate the following limit : 

`lim_(x -> 0) [(log(3 - x) - log(3 + x))/x]`

Exercise 7.6 | Q II. (5) | Page 154

Evaluate the following limit : 

`lim_(x -> 0)[(4x + 1)/(1 - 4x)]^(1/x)`

Exercise 7.6 | Q II. (6) | Page 154

Evaluate the following limit : 

`lim_(x -> 0) [(5 + 7x)/(5 - 3x)]^(1/(3x))`

Exercise 7.6 | Q III. (1) | Page 154

Evaluate the following limit : 

`lim_(x ->0) [("a"^x - "b"^x)/(sin(4x) - sin(2x))]`

Exercise 7.6 | Q III. (2) | Page 154

Evaluate the following limit : 

`lim_(x -> 0)[(2^x - 1)^3/((3^x - 1)*sinx*log(1 + x))]`

Exercise 7.6 | Q III. (3) | Page 154

Evaluate the following limit : 

`lim_(x -> 0)[(15^x - 5^x - 3^x + 1)/(x*sinx)]`

Exercise 7.6 | Q III. (4) | Page 154

Evaluate the following limit : 

`lim_(x -> 0) [((25)^x - 2(5)^x + 1)/(x*sinx)]`

Exercise 7.6 | Q III. (5) | Page 154

Evaluate the following limit :

`lim_(x -> 0) [((49)^x - 2(35)^x + (25)^x)/(sinx* log(1 + 2x))]`

Exercise 7.7 [Page 157]

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board Chapter 7 Limits Exercise 7.7 [Page 157]

Exercise 7.7 | Q I. (1) | Page 157

Evaluate the following :

`lim_(x -> ∞) [("a"x^3 + "b"x^2 + "c"x + "d")/("e"x^3 + "f"x^2 + "g"x + "h")]`

Exercise 7.7 | Q I. (2) | Page 157

Evaluate the following :

`lim_(x -> ∞) [(x^3 + 3x + 2)/((x + 4)(x - 6)(x - 3))]`

Exercise 7.7 | Q I. (3) | Page 157

Evaluate the following :

`lim_(x -> ∞) [(7x^2 + 5x - 3)/(8x^2 - 2x + 7)]`

Exercise 7.7 | Q II. (1) | Page 157

Evaluate the following :

`lim_(x -> ∞) [(7x^2 + 2x - 3)/(sqrt(x^4 + x + 2))]`

Exercise 7.7 | Q II. (2) | Page 157

Evaluate the following :

`lim_(x -> ∞) [sqrt(x^2 + 4x + 16) - sqrt(x^2 + 16)]`

Exercise 7.7 | Q II. (3) | Page 157

Evaluate the following :

`lim_(x -> ∞) [sqrt(x^4 + 4x^2) - x^2]`

Exercise 7.7 | Q III. (1) | Page 157

Evaluate the following :

`lim_(x -> ∞) [((3x^2 + 4)(4x^2 - 6)(5x^2 + 2))/(4x^6 + 2x^4 - 1)]`

Exercise 7.7 | Q III. (2) | Page 157

Evaluate the following :

`lim_(x -> ∞) [((3x - 4)^3 (4x + 3)^4)/(3x + 2)^7]`

Exercise 7.7 | Q III. (3) | Page 157

Evaluate the following :

`lim_(x -> ∞) [sqrt(x)(sqrt(x + 1) - sqrt(x))]`

Exercise 7.7 | Q III. (4) | Page 157

Evaluate the following :

`lim_(x -> ∞) [((2x - 1)^20 (3x - 1)^30)/(2x + 1)^50]`

Exercise 7.7 | Q III. (5) | Page 157

Evaluate the following :

`lim_(x -> ∞) [(sqrt(x^2 + 5) - sqrt(x^2 - 3))/(sqrt(x^2 + 3) - sqrt(x^2 + 1))]`

Miscellaneous Exercise 7 [Pages 158 - 159]

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board Chapter 7 Limits Miscellaneous Exercise 7 [Pages 158 - 159]

Miscellaneous Exercise 7 | Q I. (1) | Page 158

Select the correct answer from the given alternatives.

`lim_(x -> 2) ((x^4 - 16)/(x^2 - 5x + 6))` =

  • 23

  • 32

  • – 32

  • – 16

Miscellaneous Exercise 7 | Q I. (2) | Page 158

Select the correct answer from the given alternatives.

`lim_(x -> -2)((x^7 + 128)/(x^3 + 8))` =

  • `56/3`

  • `112/3`

  • `121/3`

  • `28/3`

Miscellaneous Exercise 7 | Q I. (3) | Page 158

Select the correct answer from the given alternatives.

`lim_(x -> 3) (1/(x^2 - 11x + 24) + 1/(x^2 - x - 6))` = 

  • `-2/25`

  • `2/25`

  • `7/25`

  • `-7/25`

Miscellaneous Exercise 7 | Q I. (4) | Page 158

Select the correct answer from the given alternatives.

`lim_(x -> 5) ((sqrt(x + 4) - 3)/(sqrt(3x - 11) - 2))` =

  • `(-2)/9`

  • `2/7`

  • `5/9`

  • `2/9`

Miscellaneous Exercise 7 | Q I. (5) | Page 158

Select the correct answer from the given alternatives.

`lim_(x -> pi/3) ((tan^2 x - 3)/(sec^3  x - 8))` =

  • 1

  • `1/2`

  • `1/3`

  • `1/4`

Miscellaneous Exercise 7 | Q I. (6) | Page 158

Select the correct answer from the given alternatives.

`lim_(x -> 0) ((5sinx - xcosx)/(2tanx - 3x^2))` =

  • 0

  • 1

  • 2

  • 3

Miscellaneous Exercise 7 | Q I. (7) | Page 158

Select the correct answer from the given alternatives.

`lim_(x -> pi/2) [(3cos x + cos 3x)/(2x - pi)^3]` =

  • `3/2`

  • `1/2`

  • `-1/2`

  • `1/4`

Miscellaneous Exercise 7 | Q I. (8) | Page 158

Select the correct answer from the given alternatives.

`lim_(x -> 0) ((15^x - 3^x - 5^x + 1)/sin^2x)` =

  • log 15

  • log 3 + log 5

  • log 3 . log 5

  • 3 log 5

Miscellaneous Exercise 7 | Q I. (9) | Page 158

Select the correct answer from the given alternatives.

`lim_(x -> 0) ((3 + 5x)/(3 - 4x))^(1/x)` =

  • e3 

  • e6 

  • e9 

  • e-3  

Miscellaneous Exercise 7 | Q I. (10) | Page 158

Select the correct answer from the given alternatives.

`lim_(x -> 0) [(log(5 + x) - log(5 - x))/sinx]` =

  • `3/2`

  • `-5/2`

  • `-1/2`

  • `2/5`

Miscellaneous Exercise 7 | Q I. (11) | Page 158

Select the correct answer from the given alternatives.

`lim_(x -> pi/2) ((3^(cosx) - 1)/(pi/2 - x))` =

  • 1

  • log 3

  • `3^(pi/2)`

  • 3 log 3

Miscellaneous Exercise 7 | Q I. (12) | Page 158

Select the correct answer from the given alternatives.

`lim_(x -> 0) [(x*log(1 + 3x))/("e"^(3x) - 1)^2]` =

  • `1/"e"^9`

  • `1/"e"^3`

  • `1/9`

  • `1/3`

Miscellaneous Exercise 7 | Q I. (13) | Page 158

Select the correct answer from the given alternatives.

`lim_(x -> 0)[(3^(sinx) - 1)^3/((3^x - 1)*tanx*log(1 + x))]` =

  • 3 log 3

  • 2 log 3

  • (log 3)2 

  • (log 3)3 

Miscellaneous Exercise 7 | Q I. (14) | Page 158

Select the correct answer from the given alternatives.

`lim_(x -> 3) [(5^(x - 3) - 4^(x - 3))/(sin(x - 3))]` =

  • log 5 – 4

  • `log  5/4`

  • `log5/log4`

  • `log5/4`

Miscellaneous Exercise 7 | Q I. (15) | Page 159

Select the correct answer from the given alternatives.

`lim_(x -> ∞) [((2x + 3)^7 (x - 5)^3)/(2x - 5)^10]` =

  • `3/8`

  • `1/8`

  • `1/6`

  • `1/4`

Miscellaneous Exercise 7 [Page 159]

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board Chapter 7 Limits Miscellaneous Exercise 7 [Page 159]

Miscellaneous Exercise 7 | Q II. (1) | Page 159

Evaluate the following :

`lim_(x -> 0)[((1 - x)^5 - 1)/((1 - x)^3 - 1)]`

Miscellaneous Exercise 7 | Q II. (2) | Page 159

Evaluate the following :

`lim_(x -> 0)[x]` ([*] is a greatest integer function.)

Miscellaneous Exercise 7 | Q II. (3) | Page 159

Evaluate the following :

If f(r) = πr2 then find `lim_("h" -> 0) [("f"("r" + "h") - "f"("r"))/"h"]`

Miscellaneous Exercise 7 | Q II. (4) | Page 159

Evaluate the following :

`lim_(x -> 0)[x/(|x| + x^2)]`

Miscellaneous Exercise 7 | Q II. (5) | Page 159

Evaluate the following :

Find the limit of the function, if it exists, at x = 1

f(x) = `{(7 - 4x, "for", x < 1),(x^2 + 2, "for", x ≥ 1):}`

Miscellaneous Exercise 7 | Q II. (6) | Page 159

Evaluate the following :

Given that 7x ≤ f(x) ≤ 3x2 – 6 for all x. Determine the value of `lim_(x -> 3) "f"(x)`

Miscellaneous Exercise 7 | Q II. (7) | Page 159

Evaluate the following :

`lim_(x -> 0)[(secx^2 - 1)/x^4]`

Miscellaneous Exercise 7 | Q II. (8) | Page 159

Evaluate the following :

`lim_(x -> 0)[("e"^x + "e"^-x - 2)/(x*tanx)]`

Miscellaneous Exercise 7 | Q II. (9) | Page 159

Evaluate the following :

`lim_(x -> 0) [(x(6^x - 3^x))/(cos (6x) - cos (4x))]`

Miscellaneous Exercise 7 | Q II. (10) | Page 159

Evaluate the following :

`lim_(x -> 0) [("a"^(3x) - "a"^(2x) - "a"^x + 1)/(x*tanx)]`

Miscellaneous Exercise 7 | Q II. (11) | Page 159

Evaluate the following :

`lim_(x -> "a") [(sinx - sin"a")/(x - "a")]`

Miscellaneous Exercise 7 | Q II. (12) | Page 159

Evaluate the following :

`lim_(x -> 2) [(logx - log2)/(x - 2)]`

Miscellaneous Exercise 7 | Q II. (13) | Page 159

Evaluate the following :

`lim_(x -> 1) [("ab"^x - "a"^x"b")/(x^2 - 1)]`

Miscellaneous Exercise 7 | Q II. (14) | Page 159

Evaluate the following : 

`lim_(x -> 0) [((5^x - 1)^2)/((2^x - 1)log(1 + x))]`

Miscellaneous Exercise 7 | Q II. (15) | Page 159

Evaluate the following : 

`lim_(x -> ∞) [((2x + 1)^2*(7x - 3)^3)/(5x + 2)^5]`

Miscellaneous Exercise 7 | Q II. (16) | Page 159

Evaluate the following :

`lim_(x -> "a") [(x cos "a" - "a" cos x)/(x - "a")]`

Miscellaneous Exercise 7 | Q II. (17) | Page 159

Evaluate the following :

`lim_(x -> pi/4) [(sinx - cosx)^2/(sqrt(2) - sinx - cosx)]`

Miscellaneous Exercise 7 | Q II. (18) | Page 159

Evaluate the following :

`lim_(x -> 1) [(2^(2x - 2) - 2^x + 1)/(sin^2 (x - 1))]`

Miscellaneous Exercise 7 | Q II. (19) | Page 159

Evaluate the following :

`lim_(x -> 1) [(4^(x -  1) - 2^x + 1)/(x - 1)^2]`

Miscellaneous Exercise 7 | Q II. (20) | Page 159

Evaluate the following :

`lim_(x -> 1) [(sqrt(x) - 1)/logx]`

Miscellaneous Exercise 7 | Q II. (21) | Page 159

Evaluate the following :

`lim_(x -> 0) [(sqrt(1 - cosx))/x]`

Miscellaneous Exercise 7 | Q II. (22) | Page 159

Evaluate the following :

`lim_(x -> 1) [(x + 3x^2 + 5x^3 + ... + (2"n" - 1)x^"n" - "n"^2)/(x - 1)]`

Miscellaneous Exercise 7 | Q II. (23) | Page 159

Evaluate the following :

`lim_(x -> 0) {1/x^12 [1 - cos(x^2/2) - cos(x^4/4) + cos(x^2/2) cos(x^4/4)]}`

Miscellaneous Exercise 7 | Q II. (24) | Page 159

Evaluate the following :

`lim_(x -> ∞) [(8x^2 + 5x + 3)/(2x^2 - 7x - 5)]^((4x + 3)/(8x - 1))`

Chapter 7: Limits

Exercise 7.1Exercise 7.2Exercise 7.3Exercise 7.4Exercise 7.5Exercise 7.6Exercise 7.7Miscellaneous Exercise 7
Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board - Shaalaa.com

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board chapter 7 - Limits

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board chapter 7 (Limits) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the Maharashtra State Board Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board chapter 7 Limits are Concept of Limits, Factorization Method, Rationalization Method, Limits of Trigonometric Functions, Substitution Method, Limits of Exponential and Logarithmic Functions, Limit at Infinity.

Using Balbharati 11th solutions Limits exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in Balbharati Solutions are important questions that can be asked in the final exam. Maximum students of Maharashtra State Board 11th prefer Balbharati Textbook Solutions to score more in exam.

Get the free view of chapter 7 Limits 11th extra questions for Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board and can use Shaalaa.com to keep it handy for your exam preparation

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