I am having trouble finding the limit for the following function: lim x → 0 f ( x ) = tan ⁡ 3 x tan ⁡ 2 x I do not want to use L'Hospital's rule.I know that tan ⁡ 3 x tan ⁡ 2 x = sin ⁡ 3 x cos ⁡ 2 x sin ⁡ 2 x cos ⁡ 3 x , but that does not help me.Can you maybe give me a hint?

Question

I am having trouble finding the limit for the following function:      lim          x      →      0        f  (  x  )  =            tan      ⁡      3      x              tan      ⁡      2      x      I do not want to use L&#039;Hospital&#039;s rule.I know that             tan      ⁡      3      x              tan      ⁡      2      x        =            sin      ⁡      3      x      cos      ⁡      2      x              sin      ⁡      2      x      cos      ⁡      3      x      , but that does not help me.Can you maybe give me a hint?

Caiden Barrett · Accepted Answer

sin      ⁡      3      x      cos      ⁡      2      x              sin      ⁡      2      x      cos      ⁡      3      x        =      3    2    .            sin      ⁡      3      x              3      x        .            2      x              sin      ⁡      2      x        .            cos      ⁡      2      x              cos      ⁡      3      x

pipantasi4 · Accepted Answer

Using   tan  ⁡  (  A  +  B  )  =            tan      ⁡      A      +      tan      ⁡      B              1      −      tan      ⁡      A      tan      ⁡      B       twice we obtain  tan  ⁡  2  x  =            2      tan      ⁡      x              1      −              tan        2            ⁡      x        ;  tan  ⁡  3  x  =            3      tan      ⁡      x      −              tan        3            ⁡      x              1      −      3              tan        2            ⁡      x      so that            tan      ⁡      2      x              tan      ⁡      3      x        =            2      (      1      −      3              tan        2            ⁡      x      )              (      3      −              tan        2            ⁡      x      )      (      1      −              tan        2            ⁡      x      )      And (having cancelled a factor of tanx) nothing vanishes and the limit is easy to find, even if you know no other limits.

I am having trouble finding the limit for the following function: <munder> <mo movablelimits

Answered question

Answer & Explanation

New Questions in Differential Calculus