Objectives: In this tutorial, we define what it means to take a limit as the variable x approaches either infinity or negative infinity. We investigate this concept from the numerical, graphical and symbolic points of view. Using this concept, we define what is meant by a horizontal asymptote.

After working through these materials, the student should be able

• to understand the definition of what it means to say that x approaches infinity;
• to become familiar with the definition of a horizontal asymptote;
• to recognize graphically a horizontal asymptote; and
• to compute symbolically limits where x approaches infinity and the resulting horizontal asymptotes.

Modules:

 Intuitive Definition. Let y = f(x) be a function. Suppose that L is a number such that whenever x is large, f(x) is close to L and suppose that f(x) can be made as close as we want to L by making x larger. Then we say that the limit of f(x) as x approaches infinity is L and we write Similarly, suppose that M is a number such that whenever x is a large negative number, f(x) is close to M and suppose that f(x) can be made as close as we want to M by making x a larger negative number. Then we say that the limit of f(x) as x approaches -infinity is M and we write In either case, we refer to the each of the lines y = L and y = M as a horizontal asymptote of the function f.

• Discussion [Using Flash]

• Javascript generated numerical evidence for horizontal asymptotes.

• Javascript generated numerical evidence for some more examples of horizontal asymptotes.

• A LiveMath notebook to be used in graphically determining horizontal asymptotes.

• Using the example in the previous LiveMath notebook as a model, we make the following definition.

 Definition. Let y = f(x) be a function. We say that the limit of f(x) as x approaches infinity is L and we write if, given e > 0, there exists N such that x > N implies |f(x) - L| < e.

• Example:

• An argument similar to that given in the previous example can be used to show the following.

 Theorem. Let n be a positive real number. Then

• We can use this theorem to calculate limits of rational functions. Suppose that f = g/h is a rational function where the degree of the polynomial g is r and the degree of the polynomial h is s. Let n be the larger of r and s. We divide each of g and h by xn and then calculate the limit.

• Examples.

• We note the following limits:

 Theorem.

• Drill problems on finding horizontal asymptotes.

• Using a graphing calculator to numerically determine horizontal asymptotes.

• Problems in using a graphing calculator to numerically determine horizontal asymptotes.