What is lim x → 0 x cos ⁡ x − sin ⁡ x x 2 sin ⁡ x ?

Question

What is       lim          x      →      0                  x      cos      ⁡      x      −      sin      ⁡      x                      x        2            sin      ⁡      x      ?

Quinn Everett · Accepted Answer

lim          x      →      0                  x      cos      ⁡      (      x      )      −      sin      ⁡      (      x      )                      x        2            sin      ⁡      (      x      )        =Apply l&#039;Hôpital&#039;s rule, to get:      d          d      x            (    x    cos    ⁡    (    x    )    −    sin    ⁡    (    x    )    )    =  −  x  sin  ⁡  (  x  )      d          d      x            (          x      2        sin    ⁡    (    x    )    )    =      x    2    cos  ⁡  (  x  )  +  2  x  sin  ⁡  (  x  )      lim          x      →      0        −            sin      ⁡      (      x      )              x      cos      ⁡      (      x      )      +      2      sin      ⁡      (      x      )        =  −      lim          x      →      0                  sin      ⁡      (      x      )              x      cos      ⁡      (      x      )      +      2      sin      ⁡      (      x      )        =Apply l&#039;Hôpital&#039;s rule again, to get:      d          d      x            (    sin    ⁡    (    x    )    )    =  cos  ⁡  (  x  )      d          d      x            (    x    cos    ⁡    (    x    )    +    2    sin    ⁡    (    x    )    )    =  3  cos  ⁡  (  x  )  −  x  sin  ⁡  (  x  )  −      lim          x      →      0        −            cos      ⁡      (      x      )              x      sin      ⁡      (      x      )      −      3      cos      ⁡      (      x      )        =      lim          x      →      0                  cos      ⁡      (      x      )              x      sin      ⁡      (      x      )      −      3      cos      ⁡      (      x      )        =            cos      ⁡      (      0      )              0      sin      ⁡      (      0      )      −      3      cos      ⁡      (      0      )        =  −      1    3

Brunton39 · Accepted Answer

Use taylor expansions  c  o  s  (  x  )  =  1  −            x      2              2      !        +            x      4              4      !        −            x      6              6      !        +  .  .  .  x  .  c  o  s  (  x  )  =  x  −            x      3              2      !        +            x      5              4      !        −            x      7              6      !        +  .  .  .  s  i  n  (  x  )  =  x  −            x      3              3      !        +            x      5              5      !        −            x      7              7      !        +  .  .  .  x  .  c  o  s  (  x  )  −  s  i  n  (  x  )  =      x    3    .  (  −      1          2      !        +      1          3      !        )  +      x    5    .  (      a    5    )  +  .  .  .      x    2    .  s  i  n  (  x  )  =      x    3    −            x      5              3      !        +            x      7              5      !        −            x      9              7      !        +  .  .  .      x    2    .  s  i  n  (  x  )  =      x    3    (  1  −            x      2              3      !        +            x      4              5      !        −            x      6              7      !        +  .  .  .  )Thus;            x      .      c      o      s      (      x      )      −      s      i      n      (      x      )                      x        2            .      s      i      n      (      x      )        =            (      −              1                  2          !                    +              1                  3          !                    )      +              x        2            .              a        5            +      .      .      .              (      1      +              x        2            .              b        5            +      .      .      .      )          ⟹        lim          x      →      0                  x      .      c      o      s      (      x      )      −      s      i      n      (      x      )                      x        2            .      s      i      n      (      x      )        =  −      1          2      !        +      1          3      !        =  −      1    3

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