Precalculus - Limits Involving Infinity


Sometimes while evaluating limits, we encounter the following two cases:

  • Case i: The independent variable increases or decreases without bounds such as limxfx and limx-fx. These limits are called limits at infinity.
  • Case ii: The value of the limit of a function fx becomes  or - at any real value a.

These are collectively termed limits involving infinity. The figure shown summarizes the above discussion.

Horizontal and Vertical Asymptotes

Horizontal Asymptotes: Consider the two cases of limits mentioned below.

  • limxfx=L: as x "approaches" fx "approaches" a real value L.
  • limx-fx=M: as x "approaches" -fx "approaches" a real value M.

In all the above-mentioned cases, we can see that the independent variable reaches  or -, the function reaches a real value L or M, respectively. Hence, the horizontal line y=L or y=M is called the horizontal asymptote of the function fx.

Vertical Asymptotes: Consider some of the cases of limits mentioned below.

  • limxafx= or -∞: as x "approaches" afx "approaches"  or -.
  • limxa-fx= or -∞: as x "approaches" a from the left, fx "approaches"  or -.
  • limxa+fx= or -: as x "approaches" a from the right, fx "approaches"  or -.

In all the above-mentioned cases, we can see that the function at x=a either reaches  or -. Hence, the vertical line x=a is called the vertical asymptote of the function fx.

In the graph shown above, y=-2 is the horizontal asymptote and x=1 is the vertical asymptote.

Solved Examples

Example 1: Evaluate limx3x-52x+3.

Solution: limx3x-52x+3





Example 2: Evaluate limx-11-1x.

Solution: limx-11-1x




Cheat Sheet

  • In a limit, if the independent variable approaches infinity or negative infinity, it is called a limit at infinity.
  • Likewise, in a limit, if the function approaches infinity or negative infinity at some real value of the independent variable, it is termed an infinite limit.
  • Collectively these two cases are called limits involving infinity.

Blunder Areas

  • The limit of a function at infinity can be a finite value as well as infinity.