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Question
Evaluate the following limit :
`lim_(x -> 0) [(x*tanx)/(1 - cosx)]`
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Solution
`lim_(x -> 0) (xtanx)/(1 - cosx)`
= `lim_(x -> 0) (xtanx)/(1 - cos x) xx (1 + cosx)/(1 + cosx)`
= `lim_(x -> 0) (xtanx(1 + cosx))/(1 - cos^2x)`
= `lim_(x -> 0) (xtanx(1 + cosx))/(sin^2x)`
= `lim_(x -> 0) ((tanx/x)(1 + cos x))/((sin^2x/x^2))` ...[∵ x → 0, x ≠ 0]
= `([lim_(x -> 0) tanx/x] xx [lim_(x -> 0) (1 + cosx)])/[lim_(x -> 0) sinx/x]^2`
= `(1 xx [1 + cos 0])/1^2`
= 1 + 1
= 2.
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