Limit of lim x → 0 sin ⁡ ( 5 x ) sin ⁡ ( 4 x )

Question

Limit of       lim          x      →      0                  sin      ⁡      (      5      x      )              sin      ⁡      (      4      x      )

Stevinivm · Accepted Answer

lim          x      →      0                  sin      ⁡      (      5      x      )              sin      ⁡      (      4      x      )        =      lim          x      →      0                  sin      ⁡      (      5      x      )      ⋅      5      x      ⋅      4      x              sin      ⁡      (      4      x      )      ⋅      5      x      ⋅      4      x        =      lim          x      →      0                  sin      ⁡      (      5      x      )      ⋅      4      x      ⋅      5      x              5      x      ⋅      sin      ⁡      (      4      x      )      ⋅      4      x        =      lim          x      →      0            (                  sin        ⁡        (        5        x        )                    5        x              ⋅                  4        x                    sin        ⁡        (        4        x        )              ⋅                  5        x                    4        x              )    =      (          lim              x        →        0                            sin        ⁡        (        5        x        )                    5        x              )    ⋅      (          lim              x        →        0                            4        x                    sin        ⁡        (        4        x        )              )    ⋅      (          lim              x        →        0                            5        x                    4        x              )    =      (          lim              x        →        0                            sin        ⁡        (        5        x        )                    5        x              )    ⋅      (          lim              x        →        0                            4        x                    sin        ⁡        (        4        x        )              )    ⋅      (          lim              x        →        0                    5      4        )    =  1  ⋅  1  ⋅      5    4    =      5    4

crossoverman9b · Accepted Answer

Alternatively, you could use L&#039;Hopital&#039;s rule:      lim          x      →      0                  sin      ⁡      (      5      x      )              sin      ⁡      (      4      x      )      =      lim          x      →      0                          d                  d          x                    sin      ⁡      (      5      x      )                      d                  d          x                    sin      ⁡      (      4      x      )      =      lim          x      →      0                  5      cos      ⁡      (      5      x      )              4      cos      ⁡      (      4      x      )        =      5    4

Limit of <munder> <mo form="prefix">lim <mrow class="MJX-TeXAtom-ORD"> x

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Answer & Explanation

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