Advertisements
Advertisements
Question
Evaluate the following :
`lim_(x -> ∞) [(sqrt(x^2 + 5) - sqrt(x^2 - 3))/(sqrt(x^2 + 3) - sqrt(x^2 + 1))]`
Solution
`lim_(x -> ∞) [(sqrt(x^2 + 5) - sqrt(x^2 - 3))/(sqrt(x^2 + 3) - sqrt(x^2 + 1))]`
= `lim_(x -> ∞) [(sqrt(x^2 + 5) - sqrt(x^2 - 3))/(sqrt(x^2 + 3) - sqrt(x^2 + 1)) xx (sqrt(x^2 + 5) + sqrt(x^2 - 3))/(sqrt(x^2 + 3) + sqrt(x^2 + 1)) xx (sqrt(x^2 + 3) + sqrt(x^2 + 1))/(sqrt(x^2 + 5) + sqrt(x^2 - 3))]`
= `lim_(x -> ∞)[((x^2 + 5) - (x^2 - 3))/((x^2 + 3) - (x^2 + 1)) xx(sqrt(x^2 + 3) + sqrt(x^2 + 1))/(sqrt(x^2 + 5) + sqrt(x^2 - 3))]`
= `lim_(x -> ∞) (8(sqrt(x^2 + 3) + sqrt(x^2 + 1)))/(2(sqrt(x^2 + 5) + sqrt(x^2 - 3))`
= `4lim_(x -> ∞) [((sqrt(x^2 + 3) + sqrt(x^2 + 1))/x)/((sqrt(x^2 + 5) + sqrt(x^2 - 3))/x)] ...[("Divide numerator and"),("denominator by" x)]`
= `4lim_(x -> ∞) [((sqrt(x^2 + 3))/x + (sqrt(x^2 + 1))/x)/(sqrt(x^2 + 5)/x + sqrt(x^2 - 3)/x)]`
= `(4lim_(x -> ∞)[sqrt((x^2 + 3)/x^2) + sqrt((x^2 + 1)/x^2)])/(lim_(x -> ∞)[sqrt((x^2 + 5)/x^2) + sqrt((x^2 - 3)/x^2)])`
= `(4(lim_(x -> ∞) sqrt(1 + 3/x^2) + lim_(x -> ∞) sqrt(1 + 1/x^2]))/(lim_(x -> ∞) sqrt(1 + 5/x^2) + lim_(x -> ∞) sqrt(1 - 3/x^2)`
= `(4(sqrt(1 + 0) + sqrt(1 + 0)))/(sqrt(1 + 0) + sqrt(1 - 0)) ...[lim_(x -> ∞) 1/x^"k" = 0, "k" > 0]`
= `(4(1 + 1))/(1 + 1)`
= 4
APPEARS IN
RELATED QUESTIONS
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]`
Select the correct answer from the given alternatives.
`lim_(x -> ∞) [((2x + 3)^7 (x - 5)^3)/(2x - 5)^10]` =
Evaluate the following :
`lim_(x -> ∞) [((2x + 1)^2*(7x - 3)^3)/(5x + 2)^5]`
Evaluate the following :
`lim_(x -> ∞) [(8x^2 + 5x + 3)/(2x^2 - 7x - 5)]^((4x + 3)/(8x - 1))`
