Evaluate the following limit :
`lim_(x -> 0)[(sqrt(x^2 + 9) - sqrt(2x^2 + 9))/(sqrt(3x^2 + 4) - sqrt(2x^2 + 4))]`
Solution
`lim_(x -> 0)(sqrt(x^2 + 9) - sqrt(2x^2 + 9))/(sqrt(3x^2 + 4) - sqrt(2x^2 + 4))`
= `lim_(x -> 0) (sqrt(x^2 + 9) - sqrt(2x^2 + 9))/(sqrt(3x^2 + 4) - sqrt(2x^2 + 4)) xx (sqrt(x^2 + 9) + sqrt(2x^2 + 9))/(sqrt(x^2 + 9) + sqrt(2x^2 + 9)) xx (sqrt(3x^2 + 4) + sqrt(2x^2 + 4))/(sqrt(3x^2 + 4) + sqrt(2x^2 + 4))`
= `lim_(x -> 0) ([(x^2 + 9) - (2x^2 + 9)][sqrt(3x^2 + 4) + sqrt(2x^2 + 4)])/([(3x^2 + 4) - (2x^2 + 4)][sqrt(x^2 + 9) + sqrt(2x^2 + 9)])`
= `lim_(x -> 0) (-x^2[sqrt(3x^2 + 4) + sqrt(2x^2 + 4)])/(x^2[sqrt(x^2 + 9) + sqrt(2x^2 + 9)]`
= `lim_(x -> 0) (-[sqrt(3x^2 + 4) + sqrt(2x^2 + 4)])/(sqrt(x^2 + 9) + sqrt(2x^2 + 9)) ...[(because x -> 0"," x ≠ 0),(therefore x^2 ≠ 0)]`
= `(-lim_(x -> 0) [sqrt(3x^2 + 4) + sqrt(2x^2 + 4)])/(lim_(x -> 0) [sqrt(x^2 + 9) + sqrt(2x^2 + 9)])`
= `(-[sqrt(0 + 4) + sqrt(0 + 4)])/(sqrt(0 + 9) + sqrt(0 + 9)`
= `- ((2 + 2))/(3 + 3)`
= `-2/3`
