Limits Involving Infinity; Asymptotes of Graphs

Infinite Limits

Infinite limits describe the behavior of a function as the input approaches a specific value and the output grows without bound. This concept is fundamental in understanding discontinuities and asymptotic behavior in calculus.

• Definition: If , the values of can be made arbitrarily large by requiring to be sufficiently close to .

• Negative Infinite Limit: means becomes arbitrarily negative as approaches .

• Vertical Asymptote: The line is a vertical asymptote of if or if any one-sided limit equals .

Example: Find .

• As approaches 0 from either side, increases rapidly.

Table Purpose: This table demonstrates how becomes extremely large as gets closer to 0, illustrating the concept of an infinite limit.

Finding Vertical Asymptotes Algebraically

Vertical asymptotes occur where the denominator of a rational function equals zero and the numerator does not cancel the zero.

• Steps:

  1. Factor both numerator and denominator.

  2. Reduce the fraction, if possible.

  3. Set the denominator equal to zero and solve for .

• Example: has vertical asymptotes at and a removable discontinuity at .

Finite Limits as

Definition of a Limit at Infinity

Limits at infinity describe the behavior of a function as the input grows without bound in either the positive or negative direction.

• Definition: means approaches as becomes very large.

• Horizontal Asymptote: The line is a horizontal asymptote if .

Example:

General Rule: for any real numbers , where .

Finding Infinite Limits Algebraically

When evaluating limits at infinity for rational functions, focus on the highest power terms in the numerator and denominator.

• If the highest power in the numerator () equals the highest power in the denominator ():

• If :

• If :

Infinite Limits at Infinity

Infinite limits at infinity occur when the function grows without bound as approaches infinity or negative infinity.

• Notation:

• Example:

Oblique (Slant) Asymptotes

Definition and Identification

Oblique asymptotes occur when the degree of the numerator is exactly one greater than the degree of the denominator in a rational function.

• Form:

• Finding: Divide the numerator by the denominator to express as a linear function plus a remainder that vanishes as .

• Example: For , perform polynomial division to find the oblique asymptote.

Summary: Infinite Limit vs Limit at Infinity

• Infinite Limits (Vertical Asymptotes): ; asymptote at .

• Limit at Infinity (Horizontal Asymptotes): ; asymptote at .

• Oblique Asymptotes: Occur when numerator degree is one higher than denominator.

Worked Examples and Graphical Analysis

Example: Analyzing Limits and Asymptotes from a Graph

The graph below illustrates a rational function with both vertical and horizontal asymptotes, as well as removable discontinuities.

Graph Purpose: This graph is used to visually identify vertical and horizontal asymptotes, and to analyze the behavior of the function near these points.

• Vertical Asymptotes: Where the function approaches infinity as approaches a specific value.

• Horizontal Asymptotes: Where the function approaches a constant value as approaches infinity or negative infinity.

• Removable Discontinuities: Points where the function is not defined but can be made continuous by redefining the value.

Summary Table: Behavior of Near Zero

This table shows how increases rapidly as approaches zero, illustrating the concept of infinite limits.

Key Takeaways

• Infinite limits describe unbounded behavior near specific points (vertical asymptotes).

• Limits at infinity describe the behavior as grows large (horizontal asymptotes).

• Oblique asymptotes occur when the numerator's degree is one higher than the denominator's.

• Algebraic techniques and graphical analysis are essential for identifying asymptotes and understanding function behavior.