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Question
Evaluate the following :
`lim_(x -> 0)[(secx^2 - 1)/x^4]`
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Solution
`lim_(x -> 0)(secx^2 - 1)/x^4`
Put x2 = y
As x → 0, x2 → 0
∴ y → 0
∴ Required limit
= `lim_(y -> 0) (sec y - 1)/(y^2)`
= `lim_(y -> 0) (sec y - 1)/(y^2) xx (sec y + 1)/(sec y + 1)`
= `lim_(y -> 0) (sec^2 y - 1)/(y^2(sec y + 1))`
= `lim_(y -> 0) (tan^2 y)/(y^2(sec y + 1))`
= `lim_(y -> 0) (tan^2 y)/(y^2) xx 1/(sec y + 1)`
= `lim_(y -> 0) ((tan^2 y)/y^2) xx lim_(y -> 0) 1/(sec y + 1)`
= `(1)^2 xx 1/(1 + 1)`
= `1/2`
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