If f(x)=limn→∞ ∑r=0n tan⁡x2r+1+tan3x2r+11−tan2⁡x2r+1 then limx→0 f(x)x is equal to

If f(x)=limnr=0ntanx2r+1+tan3x2r+11tan2x2r+1 then limx0f(x)x is equal to

  1. A

    1

  2. B

    0

  3. C

    -1

  4. D

    None of these

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    Solution:

    Let αr=x2r+1,r=0,1,2,,n. Then,

           tanx2r+1+tan3x2r+11tan2x2r+1=tanα+tan3αr1tan2αr=tanαr1+tan2αr1tan2αr

         =tanαrcos2αr=sinαrcosαrcos2αr=sin2αrαrcosαrcos2αr=tan2αrtanαr

     f(x)=limnr=0ntan2αrtanαr

     f(x)=limnr=0ntanx2rtanx2r+1 f(x)=limntanxtanx2n+1=tanx

    Hence, limx0f(x)x=limx0tanxx=1

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