Limits at Plus and Minus Infinity and Horizontal Asymptotes

Definitions

• Limit at Infinity:

  • If ( f(x) ) becomes arbitrarily close to a number ( P ) for all sufficiently large positive ( x ), ( P ) is the limit of ( f(x) ) as ( x ) approaches infinity.
  • Notation: ( \lim_{x \to \infty} f(x) = P ).

• Limit at Negative Infinity:

  • If ( f(x) ) becomes arbitrarily close to a number ( Q ) for all sufficiently large negative ( x ), ( Q ) is the limit of ( f(x) ) as ( x ) approaches negative infinity.
  • Notation: ( \lim_{x \to -\infty} f(x) = Q ).

Horizontal Asymptotes

• If ( \lim_{x \to \infty} f(x) = P ), then the line ( y = P ) is a rightward horizontal asymptote.
• If ( \lim_{x \to -\infty} f(x) = Q ), then the line ( y = Q ) is a leftward horizontal asymptote.

Examples

Example 1: ( f(x) = x + \frac{1}{x} )

• Graph: Y = 1 is both a rightward and leftward horizontal asymptote.
• ( \lim_{x \to \pm\infty} f(x) = 1 ).
• Explanation: ( f(x) = 1 + \frac{1}{x} ); ( \frac{1}{x} ) is small for large positive or negative ( x ).

Example 2: ( f(x) = \frac{1}{x^2 + 1} )

• Graph: ( y = 0 ) is a horizontal asymptote.
• ( \lim_{x \to \pm\infty} f(x) = 0 ).

Example 3: ( f(x) = \frac{2x - x^2 - 2}{x^3 - 1} )

• Graph: ( y = 2 ) is both a rightward and leftward horizontal asymptote.
• ( \lim_{x \to \pm\infty} f(x) = 2 ).

Example 4: Function with Different Limits

• Function: ( f(x) = \frac{x}{\sqrt{x^2 + 1}} )
• Graph: ( y = 1 ) is a rightward horizontal asymptote; ( y = -1 ) is a leftward horizontal asymptote.
• ( \lim_{x \to \infty} f(x) = 1 ); ( \lim_{x \to -\infty} f(x) = -1 ).

Calculation of Limits

Rational Functions

• For a rational function ( f(x) = \frac{p(x)}{q(x)} ):
  • Case 1: Degree of numerator < Degree of denominator
    • ( \lim_{x \to \pm\infty} f(x) = 0 ).
  • Case 2: Degrees are equal
    • ( \lim_{x \to \pm\infty} f(x) = \frac{\text{leading coefficient of the numerator}}{\text{leading coefficient of the denominator}} ).
  • Case 3: Degree of numerator > Degree of denominator
    • Limit does not exist; ( f(x) \to \pm\infty ).

Examples

• Function: ( f(x) = \frac{3x^3 + x^2 - 5}{2x^2 - x + 1} )

  • Simplify by multiplying by ( \frac{1}{x^3} )
  • ( \lim_{x \to \pm\infty} = 3 ).

• Function: ( f(x) = \frac{x^2 + x + 3}{5x^3 - 1} )

  • ( \lim_{x \to \pm\infty} = 0 ).

Non-rational Function Example

• Function: ( f(x) = 3x - 2|x| )
  • For positive ( x ), limit is 1.
  • For negative ( x ), limit is 5.

Special Cases

• For non-rational functions where limits are involved:

  • Example: ( f(x) = \sin(x) )
    • Limit does not exist as function oscillates.

• Basic Limits

  • ( \lim_{x \to \infty} \frac{1}{x^p} = 0 ) for ( p > 0 ).

Conclusion

• Understanding limits at infinity and horizontal asymptotes is crucial for analyzing the end behavior of functions and determining their asymptotic behavior.