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Question
Evaluate the following limit :
`lim_(x -> pi/6) [(2 - "cosec"x)/(cot^2x - 3)]`
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Solution
`lim_(x -> pi/6) [(2 - "cosec"x)/(cot^2x - 3)]`
= `lim_(x -> pi/6) (2 - "cosec"x)/("cosec"^2x - 1 - 3)`
= `lim_(x -> pi/6) (2 - "cosec"x)/("cosec"^2x - 4)`
= `lim_(x -> pi/6) (-("cosec"x - 2))/(("cosec"x - 2)("cosec"x + 2))`
= `lim_(x -> pi/6) (-1)/(("cosec"x + 2)) ...[(because x -> pi/6"," therefore x ≠ pi/6"," therefore "cosec" pi/6","),(therefore "cosec"x ≠ 2"," therefore "cosec"x - 2 ≠ 0)]`
= `(-1)/("cosec"(pi/6) + 2)`
= `(-1)/(2 + 2)`
= `(-1)/4`
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