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.
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]
- 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.
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.
- An argument similar to that given in the previous example can be used
to show the following.
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.
- We note the following limits:
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.