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

Evaluate the following limit:

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

Evaluate the following limit:

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

Evaluate the following limit:

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

Evaluate the following limit:

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

Evaluate the following limit:

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

Evaluate the following limit:

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

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

Evaluate the following limit :

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

Evaluate the following limit :

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

Evaluate the following limit : 

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

Evaluate the following limit :

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

Evaluate the following limit :

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

Evaluate the following limit :

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

Evaluate the following limit :

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

Evaluate the following limit :

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

Evaluate the following limit :

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

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

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

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

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

Evaluate the following limit :

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

Evaluate the following limit :

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

Evaluate the following limit :

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

Evaluate the following limit :

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

Evaluate the following limit :

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

Evaluate the following limit :

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

Evaluate the following limit :

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

Evaluate the following limit :

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

Evaluate the following limit :

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

Evaluate the following limit :

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

Evaluate the following limit :

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

Evaluate the following limit :

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

Evaluate the following limit :

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

Evaluate the following limit :

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

Evaluate the following limit :

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

Evaluate the following limit :

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

Evaluate the following limit :

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

Evaluate the following limit :

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

Evaluate the following limit :

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

Evaluate the following limit :

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

Evaluate the following limit :

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

Evaluate the following limit :

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

Evaluate the following limit :

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

Evaluate the following limit :

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

Evaluate the following limit :

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

Evaluate the following limit :

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

Evaluate the Following limit :

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

Evaluate the Following limit :

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

Evaluate the Following limit :

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

Evaluate the Following limit :

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

Evaluate the following limit :

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

Evaluate the following limit :

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

Evaluate the following limit :

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

Evaluate the following limit :

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

Evaluate the following limit :

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

Evaluate the following limit :

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

Evaluate the following limit :

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

Evaluate the following limit :

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

Evaluate the following limit :

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

Evaluate the following limit :

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

Evaluate the following limit :

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

Evaluate the following :

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

Evaluate the following :

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

Evaluate the following :

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

Evaluate the following :

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

Evaluate the following :

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

Evaluate the following :

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

Evaluate the following :

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

Evaluate the following :

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

Evaluate the following :

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

Evaluate the following :

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

Evaluate the following limit : 

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

Evaluate the following limit : 

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

Evaluate the following limit : 

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

Evaluate the following limit : 

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

Evaluate the following limit : 

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

Evaluate the following limit : 

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

Evaluate the following limit : 

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

Evaluate the following limit : 

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

Evaluate the following limit : 

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

Evaluate the following limit : 

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

Evaluate the following limit : 

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

Evaluate the following limit : 

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

Evaluate the following limit : 

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

Evaluate the following limit : 

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

Evaluate the following limit : 

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

Evaluate the following limit :

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

Evaluate the following :

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

Evaluate the following :

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

Evaluate the following :

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

Evaluate the following :

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

Evaluate the following :

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

Evaluate the following :

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

Evaluate the following :

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

Evaluate the following :

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

Evaluate the following :

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

Evaluate the following :

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

Evaluate the following :

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

Select the correct answer from the given alternatives.

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

Select the correct answer from the given alternatives.

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

Select the correct answer from the given alternatives.

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

Select the correct answer from the given alternatives.

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

Select the correct answer from the given alternatives.

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

Select the correct answer from the given alternatives.

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

Select the correct answer from the given alternatives.

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

Select the correct answer from the given alternatives.

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

Select the correct answer from the given alternatives.

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

Select the correct answer from the given alternatives.

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

Select the correct answer from the given alternatives.

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

Select the correct answer from the given alternatives.

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

Select the correct answer from the given alternatives.

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

Select the correct answer from the given alternatives.

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

Select the correct answer from the given alternatives.

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

Evaluate the following :

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

Evaluate the following :

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

Evaluate the following :

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

Evaluate the following :

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

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):}`

Evaluate the following :

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

Evaluate the following :

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

Evaluate the following :

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

Evaluate the following :

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

Evaluate the following :

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

Evaluate the following :

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

Evaluate the following :

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

Evaluate the following :

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

Evaluate the following : 

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

Evaluate the following : 

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

Evaluate the following :

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

Evaluate the following :

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

Evaluate the following :

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

Evaluate the following :

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

Evaluate the following :

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

Evaluate the following :

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

Evaluate the following :

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

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)]}`

Evaluate the following :

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