Limits and Continuity

Infinite Limits

Infinite limits occur when the value of a function increases or decreases without bound as the input approaches a particular point. This concept is fundamental in calculus, especially when analyzing the behavior of functions near points of discontinuity or singularity.

• 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:

• This means we can make the values of f(x) arbitrarily large (or small) by taking x sufficiently close to a, but not equal to a.

• Key Point: Infinite limits describe the behavior of functions near points where they diverge to positive or negative infinity.

• Example: The function as diverges to infinity or negative infinity depending on the direction of approach.

One-Sided Limits

One-sided limits analyze the behavior of a function as the input approaches a point from one side only (left or right). This is useful for functions that behave differently on either side of a point.

• Left-hand limit:

• Right-hand limit:

• Application: These limits help describe discontinuities and asymptotic behavior.

Examples of Infinite Limits

Several classic examples illustrate infinite limits:

• Example 1:

• Example 2:

• General Case: for all even integers n.

• Example 3:

• Special Case: (not infinite, but important for comparison).

Additional info: Infinite limits are closely related to vertical asymptotes, where the function grows without bound near a specific value of x.