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
