Limits Involving Infinity

Introduction to Limits at Infinity

In calculus, the concept of a limit is extended to include cases where the independent variable approaches infinity or where the function itself approaches infinity. This is essential for understanding the behavior of functions as inputs grow large or near points where the function becomes unbounded.

• Limit at Infinity: The notation means that as increases without bound, approaches the real number .

• Infinite Limit: means that as approaches , increases without bound.

• Horizontal Asymptote: If or , then and are horizontal asymptotes of .

Graphical Interpretation

Graphs can illustrate limits involving infinity by showing how functions behave as approaches infinity or as approaches points where the function becomes unbounded.

• Vertical Asymptote: A line is a vertical asymptote if approaches infinity as approaches from either side.

• Horizontal Asymptote: A line is a horizontal asymptote if approaches as approaches infinity.

Calculating Limits at Infinity

Rules for Rational Functions

For rational functions, limits at infinity depend on the degrees of the numerator and denominator.

• If degree of numerator < degree of denominator:

• If degree of numerator = degree of denominator:

• If degree of numerator > degree of denominator: or (depending on sign)

Example 1

Find

• Divide numerator and denominator by :

• As , and

• Result:

Example 2

Find

• Divide inside the square root by :

• As , and

• Result:

Horizontal Asymptotes

A function can have zero, one, or two horizontal asymptotes, but never more than two. Horizontal asymptotes are determined by the limits as and .

• Rule: The lines and are horizontal asymptotes if these limits exist.

• Example: has horizontal asymptotes at (as ) and (as ).

• Example: has horizontal asymptotes at (as ) and (as ).

Special Cases

• No Horizontal Asymptote: If the limit does not exist as , there is no horizontal asymptote.

• Two Horizontal Asymptotes: Some functions, such as those involving square roots, can have two distinct horizontal asymptotes.

• Three Horizontal Asymptotes: Not possible for a function; a graph with three horizontal lines fails the vertical line test and is not a function.

Vertical Asymptotes

Definition and Identification

A vertical asymptote occurs at if the function approaches infinity (or negative infinity) as approaches from either side.

• Rule: is a vertical asymptote of if or , or or .

• Example: has vertical asymptotes at and .

Determining the Sign of Infinite Limits

To determine whether the limit approaches or , substitute values close to the asymptote from the left and right.

• For , as , is negative, so .

• As , is positive, so .

Limits Involving Transcendental Functions

Exponential and Trigonometric Functions

• Exponential Function: ;

• Arctangent Function: ;

Summary Table: Horizontal and Vertical Asymptotes

Examples and Applications

Example: Rational Function

• Find the horizontal asymptotes of

• Divide numerator and denominator by :

• (since )

Example: Square Root Function

• Find the horizontal asymptotes of

• As , ; as ,

Example: Vertical Asymptote

• Find the vertical asymptotes of

• Set denominator to zero:

• Check limits from left and right to determine sign ( or )

Key Takeaways

• Limits involving infinity help describe the end behavior of functions.

• Horizontal asymptotes are found by evaluating limits as and .

• Vertical asymptotes occur where the function becomes unbounded as approaches a finite value.

• Rational and transcendental functions exhibit characteristic asymptotic behavior.

Additional info: Some context and examples were inferred from partial notes and standard calculus knowledge to ensure completeness and clarity.