Limits and Continuity

Infinite Limits

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 are fundamental in understanding 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).

• Mathematical Notation:

• Interpretation: This means that as x approaches a, f(x) increases or decreases without bound.

Examples of Infinite Limits

• Example 1:

  • As x approaches 2, the denominator approaches zero, causing the function to grow without bound.

• Example 2:

  • As x approaches 0, the denominator becomes very small, and the function increases without bound.

General Property of Infinite Limits

• For any real number a and positive integer n:

Special Cases and Observations

• Example 3:

• Observation: (This is a standard limit, not infinite, but often discussed in the context of limits approaching zero.)

Additional info: Infinite limits are closely related to the concept of vertical asymptotes in graphing functions. When a function approaches infinity as x approaches a certain value, the graph of the function will have a vertical asymptote at that value.