asymptotes of x+1/x+e^(-x^2) — step-by-step solution

Let's find the asymptotes of

x + \frac{1}{x} + e^{- x^{2}}

, for real

x

The function is defined on

ℝ \ {0}

: a vertical asymptote can only be at a finite bound of this domain

x

approaches

0

from the left, the function tends to

-∞

; from the right, to

+∞

The function tends to infinity: the line

x = 0

is a vertical asymptote

x

tends to -∞ or to +∞,

\frac{f (x)}{x}

tends to

1

and

f (x) - x

tends to

0

: the line

y = x

is a slant asymptote

▶Asymptotes:

{x = 0, y = x}

🖼/draw?f1=x+%2B+1%2Fx+%2B+e%5E%28-x%5E2%29&v=x&x2=0&f3=x&b=-10.0&e=10.0&min=-12.0&max=12.0&dash=2,3