Objectives: In this tutorial, we define what it means to have a limit of f(x) approach 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 vertical asymptote.

After working through these materials, the student should be able

• to understand the definition of what it means to say that a limit is infinity;
• to become familiar with the definition of a vertical asymptote;
• to recognize graphically a vertical asymptote; and
• to compute symbolically limits which approach infinity and the resulting vertical asymptotes.

Modules:

 Intuitive Definition. Let f be a function which is defined on some open interval containing a except possibly at x = a. We write if f(x) grows arbitrarily large by choosing x sufficiently close to a. We write if f(x) grows arbitrarily large in a negative sense by choosing x sufficiently close to a. Similarly, we can define the one-sided limits:    In any of these six cases, we say that the line x = a is a vertical asymptote of f.

• Discussion [Using Flash]

• Javascript generated numerical evidence for infinite limits.

• A LiveMath notebook illustrating vertical asymptotes.

 Theorem. Suppose that f is a function that is defined on some open interval containing a, then the limit of f(x) as x approaches a is infinity if, given any real number N, there exists d > 0 such that 0 < |x - 0| < d implies f(x) > N. The the limit of f(x) as x approaches a is negative infinity if, given any real number N, there exists d > 0 such that 0 < |x - 0| < d implies f(x) < N. Analogous definitions can be made for left-hand and right-hand limits.

• Example:

• Discussion [Using Flash]

• The following theorem provides a way to calculate vertical asymptotes symbolically.

 Theorem. Suppose that is a function such that Suppose that there is an open interval I containing a such that for x in I: if f(x) > 0 for x > a, then if f(x) < 0 for x > a, then if f(x) > 0 for x < a, then if f(x) < 0 for x < a, then • Examples:
• Discussion.

• Discussion.

• Drill problems on finding vertical asymptotes.

• Using a graphing calculator to numerically determine vertical asymptotes.

• Using a graphing calculator to determine the roots and the vertical asymptotes of a rational function.