Asymptotes of arctan ⁡ 2 xMy book tells me the horizontal asymptotes of arctan ⁡ 2 x is either at positive or negative π 2 , yet the vertical asymptotes of tan ⁡ 2 x occurs at positive or negative x = π 4 , so obviously the horizontal asymptotes of arctan ⁡ 2 x should be at either positive or negative y = π 4 .Can someone tell me what I'm doing wrong?

Question

Asymptotes of   arctan  ⁡  2  xMy book tells me the horizontal asymptotes of   arctan  ⁡  2  x is either at positive or negative       π    2  , yet the vertical asymptotes of   tan  ⁡  2  x occurs at positive or negative   x  =      π    4  , so obviously the horizontal asymptotes of   arctan  ⁡  2  x should be at either positive or negative   y  =      π    4  .Can someone tell me what I&#039;m doing wrong?

Mckayla Carlson · Accepted Answer

lim          x      →      ∞        arctan  ⁡  (  2  x  )  =  π      /    2        lim          x      →      −      ∞        arctan  ⁡  (  2  x  )  =  −  π      /    2Notice that       1    2    tan  ⁡  (  arctan  ⁡  (  2  x  )  )  =  x

Chad Baldwin · Accepted Answer

The functions   x  ↦  arctan  ⁡  2  x and (some restriction of)   x  ↦  tan  ⁡  2  x are not inverses: The inverse of   x  ↦  arctan  ⁡  2  x is  x  ↦            1      2        tan  ⁡  x  ,    −            π      2        &amp;lt;  x  &amp;lt;            π      2        .

My book tells me the horizontal asymptotes of arctan 2x is either at positive or negative pi/2, yet the vertical asymptotes of tan 2x occurs at positive or negative x=pi/4, so obviously the horizontal asymptotes of arctan2x should be at either positive or negative y=pi/4.

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