2.2 Limits Involving Infinity
Horizontal Asymptote
The line 𝑦 = 𝑏 is a horizontal asymptote of the graph of a
function 𝑦 = 𝑓(𝑥) if either
lim 𝑓(𝑥) = 𝑏 𝑜𝑟 lim 𝑓(𝑥) = 𝑏
𝑥→∞
𝑥→−∞
Because this is the definition of a horizontal asymptote, this is the only criterion needed to determine if
there is a horizontal asymptote.
Vertical Asymptote
The line 𝑥 = 𝑎 is a vertical asymptote of the graph of a function
𝑦 = 𝑓(𝑥) if either
lim+ 𝑓(𝑥) = ±∞ 𝑜𝑟 lim− 𝑓(𝑥) = ±∞
𝑥→𝑎
𝑥→𝑎
Note that a limit becomes ±∞ when a finite, non-zero number is divided by (a number approaching) zero.
End Behavior Models
Sometimes if we have a complicated function, it’s helpful to have a simple function that models (acts
like) the more complicated function. End behavior models are functions that model more complicated
functions at extreme values of x (either very positive numbers or very negative numbers).
Finding End Behavior Models
To find a function g that is an end behavior model for f, look at
lim 𝑓(𝑥)
𝑥→±∞
and see what parts of the function become insignificant.
example: If 𝑓(𝑥) = 𝑥 + 𝑒 −𝑥 , we can look at:
Which leads to:
This shows us that when 𝑥 → ∞, 𝑒 −𝑥 becomes
insignificant. Only the x is important. Therefore, the
right end behavior model is:
lim (𝑥 + 𝑒 −𝑥 )
𝑥→∞
lim 𝑥 + lim 𝑒 −𝑥
𝑥→∞
1
lim 𝑥 + lim 𝑥
𝑥→∞
𝑥→∞ 𝑒
lim 𝑥 + 0
𝑥→∞
𝑥→∞
𝑔(𝑥) = 𝑥
End Behavior Model
The function g is:
(a) a right end behavior model for f if and only if
𝑓(𝑥)
lim
=1
𝑥→∞ 𝑔(𝑥)
(b) a left end behavior model for f if and only if
𝑓(𝑥)
lim
=1
𝑥→−∞ 𝑔(𝑥)
The above rules are used to show that a simple function ( g (x) ) is an end behavior model of a more
complicated function ( f (x) ).