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Question
Evaluate the following limit.
`lim_(x -> (pi)/2) (tan 2x)/(x - pi/2)`
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Solution
Substituting x = `π/2 + h` in `lim_(x → π/2) (tan2x)/(x - π/2)`,
= `lim_(h → 0) (tan 2(π/2 + h))/h`
= `lim_(h → 0) (tan(π + 2h))/h`
= `lim_(h → 0) (tan 2h)/h`
= `lim_(h → 0) (sin 2h)/2h . 2/(cos 2h)`
= `lim_(h → 0) 2/(cos 2h)` ....`(∵ lim_(h → 0) (sin2h)/(2h) = 1)`
= `2/cos0`
= `2/1`
= 2
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