Finding asymptotes of a function using limitsI need to find asymptotes of a function. I didn't find any vertical asymptotes and I think there aren't any. But when trying to find a horizontal asymptote I evaluate the limit as x approaches infinity but can't successfully find it. f ( x ) = x arctan ⁡ ( x )

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Finding asymptotes of a function using limitsI need to find asymptotes of a function. I didn&#039;t find any vertical asymptotes and I think there aren&#039;t any. But when trying to find a horizontal asymptote I evaluate the limit as x approaches infinity but can&#039;t successfully find it.  f  (  x  )  =  x  arctan  ⁡  (  x  )

kilinumad · Accepted Answer

The function f has an oblique asymptote   y  =  a  x  +  b when   x  →  ∞ iff      lim          x      →      ∞                  f      (      x      )        x    =  a      lim          x      →      ∞        (  f  (  x  )  −  a  x  )  =  bSimilar conditions hold for the case   x  →  −  ∞.For   f  (  x  )  =  x  arctan  ⁡  (  x  ) we have when   x  →  +  ∞            f      (      x      )        x    =  arctan  ⁡  x  →      π    2    f  (  x  )  −      π    2    x  =  x  (  arctan  ⁡  x  −      π    2    )  =  −  x  arctan  ⁡      (          1      x        )    →  −  1Hence there is an oblique asymptote   y  =      π    2    x  −  1 at   +  ∞.What is the asymptote when   x  →  −  ∞ ?

Finding asymptotes of a function using limits I need to find asymptotes of a function. I didn't find any vertical asymptotes and I think there aren't any. But when trying to find a horizontal asymptote I evaluate the limit as x approaches infinity but can't successfully find it. f(x)=xarctan(x)

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