Limits at Infinity

Introduction to Limits at Infinity

Limits at infinity are used to analyze the behavior of functions as the variable approaches very large positive or negative values. This concept is essential for understanding end behavior of polynomials and rational functions in calculus.

• Limit at Infinity: The value that a function approaches as the input grows without bound.

• Rational Functions: Functions of the form , where both and are polynomials.

• Key Strategy: Divide numerator and denominator by the highest power of in the denominator to simplify the limit.

Evaluating Limits at Infinity for Rational Functions

To evaluate , compare the degrees of the numerator and denominator polynomials.

• Case 1: Degrees are Equal ()

  • The limit is the ratio of the leading coefficients:

  • Example:

• Case 2: Degree of Numerator < Degree of Denominator ()

  • The limit is $0$.

  • Example:

• Case 3: Degree of Numerator > Degree of Denominator ()

  • The limit is or , depending on the sign of the leading coefficients.

Properties of Exponents in Limits

• Exponent Rule:

• Application: Used to simplify terms when dividing by the highest power of .

Examples of Limits at Infinity

• Example 1:

  • Divide numerator and denominator by :

• Example 2:

  • Divide by :

Continuity of a Function

Definition of Continuity

A function is continuous at a number if:

• is defined (i.e., is in the domain of )

• exists

These three conditions must all be satisfied for continuity at a point.

Checking Continuity: Examples

• Example 1:

  • Domain:

  • At , is undefined, so is discontinuous at .

  • Since is not defined, discontinuity occurs at .

• Example 2:

  • At ,

  • Since , is discontinuous at .

• Example 3:

  • At , is undefined.

  • Discontinuity at .

Summary Table: Continuity Conditions

Key Formulas and Properties

• Limit of Rational Function at Infinity:

  • If and are polynomials of degree and respectively:

• Exponent Rule:

• Continuity at a Point:

Additional info:

• These notes cover foundational calculus concepts relevant for first-year college students, including limits at infinity and continuity, which are essential for understanding further topics such as derivatives and integrals.