Limits & Continuity

Infinite Limits

Infinite limits occur when the value of a function increases or decreases without bound as the input approaches a certain point. This concept is fundamental in understanding the behavior of functions near points where they are not defined or where they exhibit vertical asymptotes.

• 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 infinite if the values of f(x) become arbitrarily large (positive or negative) as x gets closer to a but not equal to a.

• Notation: or

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

One-Sided Infinite Limits

Sometimes, the behavior of a function as x approaches a point from the left or right can be different. One-sided limits help describe this behavior.

• Left-hand limit:

• Right-hand limit:

• Example: ,

Examples of Infinite Limits

Infinite limits often occur in rational functions where the denominator approaches zero.

• 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 approaches zero, and the function increases without bound.

• General Case: for any positive integer n.

Special Cases and Other Limits

• Example:

• Oscillating Limit: This limit does not tend to infinity but is a classic example of a limit that exists and is finite.

Summary Table: Infinite Limits

Additional info: These notes cover the concept of infinite limits, one-sided limits, and provide graphical and algebraic examples relevant to Precalculus. The included image directly illustrates the behavior of functions near vertical asymptotes, reinforcing the explanation of infinite limits.