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Question
Evaluate the following limit :
`lim_(y -> 0)[(sqrt(1 - y^2) - sqrt(1 + y^2))/y^2]`
Solution
`lim_(y -> 0)(sqrt(1 - y^2) - sqrt(1 + y^2))/y^2`
= `lim_(y -> 0) (sqrt(1 - y^2) - sqrt(1 + y^2))/y^2 xx (sqrt(1 - y^2) + sqrt(1 + y^2))/(sqrt(1 - y^2) + sqrt(1 + y^2)`
= `lim_(y -> 0) ((1 - y^2) - (1 + y^2))/(y^2(sqrt(1 - y^2) + sqrt(1 + y^2))`
= `lim_(y -> 0) (-2y^2)/(y^2(sqrt(1 - y^2) + sqrt(1 + y^2))`
= `lim_(y -> 0) (-2)/(sqrt(1 - y^2) + sqrt(1 + y^2)) ...[(because y -> 0 "," y ≠ 0),(y^2 ≠ 0)]`
= `(lim_(y -> 0) (-2))/(lim_(y -> 0) (sqrt(1 - y^2) + sqrt(1 + y^2))`
= `(-2)/(sqrt(1 - 0) + sqrt(1 + 0))`
= – 1
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