Lecture on Asymptotes of Graphs

Introduction

• Focus on horizontal, vertical, and oblique (slant) asymptotes.
• Objectives:
  • Find horizontal and vertical asymptotes.
  • Define asymptotes, including oblique asymptotes.

Horizontal Asymptotes

• Definition: Line y = B is a horizontal asymptote if:
  • ( \lim_{{x \to \infty}} f(x) = B ) or ( \lim_{{x \to -\infty}} f(x) = B ).
• Example: For ( y = f(x) = \frac{1}{x} ):
  • Both limits as x approaches infinity and negative infinity result in 0.
  • Horizontal asymptote is ( y = 0 ).

Vertical Asymptotes

• Definition: Line x = a is a vertical asymptote if:
  • ( \lim_{{x \to a^-}} f(x) = \pm \infty ) or ( \lim_{{x \to a^+}} f(x) = \pm \infty ).
• Example: For ( y = \frac{1}{x} ), vertical asymptote at ( x = 0 ).

Example Problems

Example 1: ( y = \frac{x^2}{x^2 - 4} )

• Horizontal Asymptotes:
  • Degree of numerator equals degree of denominator.
  • Horizontal asymptote at ( y = 1 ).
• Vertical Asymptotes:
  • Function undefined at ( x = 2 ) and ( x = -2 ).
  • Vertical asymptotes at ( x = 2 ) and ( x = -2 ).

Example 2: Oblique Asymptotes

• Definition: Occurs when degree of the numerator is one more than the denominator.
• Example: ( y = \frac{x^2 - 9}{x - 1} )
  • Perform long division to find oblique asymptote.
  • Oblique asymptote at ( y = x + 1 ).
  • Vertical asymptote at ( x = 1 ).

Additional Concepts

• Removable Discontinuity:
  • May appear as a potential vertical asymptote but isn’t one.
• Example: ( f(x) = \frac{x^2 - 1}{x - 1} ) has a removable discontinuity at ( x = 1 ).

Conclusion

• Asymptotes provide insights into the behavior of a graph at extreme values.
• Understanding the types and locations can aid in graph sketching.
• Through examples, determined various asymptotes and sketched graphs based on asymptotic behavior.