Limits and 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 calculus, especially when analyzing asymptotic behavior and discontinuities.

• 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.

• Example: The function has an infinite limit as x approaches 2:

• Similarly, .

General Rule: For any real number a:

• Example:

Lateral Limits

Lateral limits (or one-sided limits) describe the behavior of a function as the input approaches a point from one side only (either from the left or the right).

• Left-hand limit:

• Right-hand limit:

These limits are useful for analyzing functions with discontinuities or vertical asymptotes.

Special Limits

• Trigonometric Example:

This limit is fundamental in calculus and is often used in proofs and applications involving derivatives.

Additional info: The notes provide graphical representations of infinite limits and lateral limits, reinforcing the concept of vertical asymptotes and the behavior of functions near points of discontinuity.