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Question
Evaluate the following limits: `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))/((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"," therefore y ≠ 0","),(therefore y^2 ≠ 0)]`
= `(-2)/(sqrt(1 - 0^2) + sqrt(1 + 0^2)`
= `(-2)/(1 + 1)`
= `(-2)/2`
= –1
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