Limits & Continuity

Infinite Limits

In calculus and precalculus, infinite limits describe the behavior of a function as the input approaches a certain value and the output increases or decreases without bound. These limits help us understand vertical asymptotes and the unbounded growth of functions near specific points.

• Definition: Let f be a function defined on both sides of a (except possibly at a itself). We say that the limit of f(x) as x approaches a is infinity if the values of f(x) can be made arbitrarily large (positive or negative) by taking x sufficiently close to a (but not equal to a).

This means that as x approaches a, f(x) increases without bound.

One-Sided Infinite Limits

We can also consider the behavior of a function as x approaches a from the left or right:

• Left-hand limit:

• Right-hand limit:

For example:

Examples of Infinite Limits

• Example 1:

• Example 2:

These graphs show that as x approaches the value where the denominator is zero, the function grows without bound, indicating a vertical asymptote.

General Property of Infinite Limits

• In general, for all real numbers a and even positive integers n.

Special Cases and Other Limits

• Example 3:

• Oscillating Limit: (This is a classic limit, not infinite, but important for understanding behavior near zero.)

Additional info: Infinite limits are closely related to the concept of vertical asymptotes in graphing rational functions. When the denominator of a rational function approaches zero and the numerator does not, the function's value increases or decreases without bound near that point.