Calculus One: Lecture 6 - Limits at Infinity and Horizontal Asymptotes

Key Concepts

• Function: ( f(x) = \tan^{-1}(x) )
  • Graph features:
    • Horizontal asymptotes at ( y = \frac{\pi}{2} ) and ( y = -\frac{\pi}{2} ).
    • Passes through the origin (0,0).
    • Increasing function.

Limits at Infinity

• Limit as ( x \to \infty ): ( \lim_{x \to \infty} \tan^{-1}(x) = \frac{\pi}{2} )
• Limit as ( x \to -\infty ): ( \lim_{x \to -\infty} \tan^{-1}(x) = -\frac{\pi}{2} )

Definition of Horizontal Asymptote

• A line ( y = L ) is a horizontal asymptote of the function ( y = f(x) ) if:
  • ( \lim_{x \to \infty} f(x) = L ) or ( \lim_{x \to -\infty} f(x) = L )

Example: Analyzing ( g(x) ) Graph

• Horizontal Asymptotes:
  • ( \lim_{x \to \infty} g(x) = 2 )
  • ( \lim_{x \to -\infty} g(x) = -1 )
• Vertical Asymptotes:
  • ( x = 0 ) and ( x = 2 )

Finding Limits at Specific Points

• Limit as ( x \to 2 ):
  • Right-hand limit: ( +\infty )
  • Left-hand limit: ( -\infty )
  • Limit does not exist because the left-hand and right-hand limits are not equal.
• Limit as ( x \to 0 ):
  • Both left and right approach ( -\infty ), thus limit is ( -\infty ).

Reciprocal Function: ( f(x) = \frac{1}{x} )

• Horizontal Asymptotes:
  • Both ( \lim_{x \to \infty} \frac{1}{x} ) and ( \lim_{x \to -\infty} \frac{1}{x} ) approach 0.
• Rules apply to any rational expression ( \frac{1}{x^r} ) where ( r ) is a positive integer.

Evaluating Limits

• Rational Expressions:
  • Divide by the highest power of ( x ) in the denominator.
  • Example calculation to show step-by-step limit evaluation.

Special Cases with Radicals

• Handling Square Roots in Denominator:
  • Divide by the absolute value, considering ( \left|x\right| ).
  • Choose the positive or negative part based on the direction of ( x ).

Examples

• Example 1: ( \lim_{x \to \infty} \frac{x + 2}{\sqrt{9x^2 + 1}} )
  • Divide by ( |x| ), result is ( \frac{1}{3} ).
• Example 2: ( \lim_{x \to -\infty} \frac{\sqrt{9x^6 - x}}{x^3} )
  • Consider absolute values, result is (-3).

Summary

• Horizontal Asymptotes: Use limits at ( \pm \infty ).
• Vertical Asymptotes: Check factors of the denominator not present in the numerator.
• Handling Limits: Show all steps; use limits to justify asymptotes.
• End Behavior: Describes what happens to the function as ( x ) approaches ( \pm \infty ).

Note: Practice with different types of limits and asymptotes to strengthen understanding. Revisit methods from pre-calculus and understand their justification through limits.