Asymptotes of a functionI have the function f ( x ) = x ⋅ arctan ⁡ ( x ) And i need to find all asymptotes, I am stuck at lim x → ∞ x ⋅ arctan ⁡ ( x ), please help me.

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Asymptotes of a functionI have the function   f  (  x  )  =  x  ⋅  arctan  ⁡  (  x  ) And i need to find all asymptotes, I am stuck at       lim          x      →      ∞        x  ⋅  arctan  ⁡  (  x  ), please help me.

gortepap6yb · Accepted Answer

There are no horizontal asymptotes because the limits at infinity do not exist (they&#039;re both positive infinities):      lim          x      →      ±      ∞        x  arctan  ⁡      x    =  +  ∞  .What about the so-called oblique asymptotes? Judging by the way the graph looks, it seems like the function might have two of them. Here&#039;s how you can find them:  a  =      lim          x      →      ±      ∞                  f      (      x      )        x    =      lim          x      →      ±      ∞                  x      arctan      ⁡              x              x    =      lim          x      →      ±      ∞            arctan    ⁡          x        =  ±      π    2    ,  b  =      lim          x      →      ±      ∞        (  f  (  x  )  −  a  x  )  =      lim          x      →      ±      ∞            (    x    arctan    ⁡          x        ±          π      2        x    )    =      lim          x      →      ±      ∞                  arctan      ⁡              x            ±              π        2                    1              /            x        .Now, to find that limit, you have to use L&#039;Hôpital&#039;s rule on it. When you&#039;re finally done (and if the limit exists), plug the values of a and b that you&#039;ve found (there are going to be two pairs of them) into this simple linear equation:  y  =  a  x  +  b  .That&#039;s going to give you the equations of your two oblique asymptotes.

Frances Pham · Accepted Answer

lim          x      →      ∞        arctan  ⁡  (  x  )  =      π    2    as   tan  ⁡  (      π    2    )  =  ∞. Then this is multiplied by x so the limit is infinity.

Asymptotes of a function I have the function f(x)=x times arctan(x) And i need to find all asymptotes, I am stuck at limx -> infinity x⋅arctan(x), please help me.

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