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)

brasocas6 2022-08-07 Answered
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 ) = x arctan ( x )
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Answers (1)

kilinumad
Answered 2022-08-08 Author has 21 answers
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 ) = b
Similar 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 ) 1
Hence there is an oblique asymptote y = π 2 x 1 at + .
What is the asymptote when x ?
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