12.4: Limits approaching infinity
Evaluating the limit of a function as it approaches infinity is really finding what the end behavior of that function is. To address end behavior, you will want to use something you already know, horizontal asymptotes. As an  gets closer to a   and , some functions reach a limit which is their horizontal asymptote.  can get arbitrarily close to the horizontal asymptote, but can never touch it. Let's look at an example.

Example:





To find the limit of the function, you need to look at the terms that will always be the largest on both the numerator and the denominator. If you were to plug in a very large number for , which term would be the largest? In this case it would be  and . Since no matter which very large number you plug in for , it will always be substantially larger for  and  than it would be for  and . you can discount the other terms when evaluating the limit at .

  is what you want to look at to evaluate the limit as . Now plug in the largest number you can think of for  and you will see that the function simplifies to become  .



Function with higher power in the denominator:


The limit is  because as  the function gets smaller and smaller and begins to approach , making  the horizontal asymptote, and the limit of the function.

Function with higher power in the numerator:


The limit as  for this function does not exist because as  the numerator will always be larger, meaning  can expand without limits.

Important to remember:
 
This is true for all functions that are not piecewise defined functions or inverse tangent functions, because both of those kinds of functions have different end behaviors.

Graphical representation of the limits (in order):









Other helpful sources:
https://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/liminfdirectory/LimitInfinity.html
https://www.mathsisfun.com/calculus/limits-infinity.html
https://www.khanacademy.org/math/differential-calculus/limits-topic/limits-infinity/v/more-limits-at-infinity