f:[0,infty)rightarrow R,f(x)=sqrt x2+xln(e^x+1) I have this function and i need to find out the asymptotes to +infty (+infinity) i calculate the horizontal ones, and they are +infty i can't calculate the limits at the obliques asymptotes

Jannek93

Jannek93

Answered question

2022-09-30

Obliques asymptotes of a function
f : [ 0 , ) R , f ( x ) = x 2 + x ln ( e x + 1 )
I have this function and i need to find out the asymptotes to + (+infinity)
i calculate the horizontal ones, and they are +
i can't calculate the limits at the obliques asymptotes

Answer & Explanation

Matthew Benton

Matthew Benton

Beginner2022-10-01Added 9 answers

If you want to find the oblique asymptote in the direction of positive infinity, then do this:
a = lim x + f ( x ) x = lim x + x 2 + x ln ( e x + 1 ) x = lim x + x 2 + x ln ( e x + 1 ) x 2 = lim x + 1 + ln ( e x + 1 ) x = 1 + 1 = 2 .
b = lim x + ( f ( x ) a x ) = lim x + ( x 2 + x ln ( e x + 1 ) 2 x ) = 0
The second limit is more difficult to calculate, but it equals 0.
Plugging all this information into the equation y=ax+b, we get:
y = 2 x .
ter3k4w8x

ter3k4w8x

Beginner2022-10-02Added 4 answers

Hint.
lim x + x 2 + x ln ( e x + 1 ) x = lim x + x 2 + x ln ( e x + 1 ) x 2 = lim x + 1 + ln ( e x + 1 ) x
Can you go on from here?

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