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Question
Evaluate: `lim_(x -> pi/6) (cot^2 x - 3)/("cosec" x - 2)`
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Solution
Given that `lim_(x -> pi/6) (cot^2 x - 3)/("cosec" x - 2)`
= `lim_(x -> pi/6) ("cosec"^2 x - 1 - 3)/("cosec" x - 2)`
= `lim_(x -> pi/6) ("cosec"^2x - 4)/("cosec" x - 2)`
= `lim_(x -> pi/6) (("cosec" x + 2)("cosec" x - 2))/(("cosec" x - 2))`
= `lim_(x -> pi/6) ("cosec" x + 2)`
Taking limit we have
= `"cosec" pi/6 + 2`
= 2 + 2
= 4
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