Finding Limits at Infinity

Understanding Limits at Infinity

• Limit as x approaches infinity for the function x squared:

  • If x = 1,000, then x squared = 1,000,000.
  • As x increases, x squared approaches infinity.

• Limit as x approaches negative infinity for x squared:

  • Squaring negative x gives a positive number.
  • Therefore, limit = positive infinity.

Cubic Functions

• Limit as x approaches infinity for positive x cubed:
  • Infinity cubed still leads to positive infinity.
• Limit as x approaches negative infinity for x cubed:
  • Negative infinity cubed results in negative infinity.

Example Problems

1. Limit as x approaches negative infinity of: 5 + 2x - x³

   • Ignore insignificant terms (5 and 2x).
   • Equivalent to limit as x approaches negative infinity of -x³.
   • Result is positive infinity.

2. Limit as x approaches negative infinity of: 3x³ - 5x⁴

   • Significant term is -5x⁴.
   • Limit = negative 5 times negative infinity to the power of 4 = negative infinity.

Rational Functions

• Limit as x approaches infinity for 1/x:

  • As x increases, value of the fraction approaches 0.
  • Conclusion: limit as x approaches infinity of 1/x is 0.

• For rational functions where the denominator's degree exceeds the numerator's:

  • The limit will always equal 0.

Example Problem

• Limit as x approaches infinity of: (5x + 2)/(7x - x²)
  • Degree of denominator (2) > degree of numerator (1) → limit = 0.

Showing Work

• Multiply top and bottom by 1/x²:
  • Resulting limit as x approaches infinity: (5/x + 2/x²) / (7/x - 1) = 0 / -1 = 0.

Same Degree in Numerator and Denominator

• Limit as x approaches infinity of: (8x² - 5x)/(4x² + 7)
  • Both degrees are the same (2).
  • Coefficients: 8/4 = 2.

Example Problem

• Limit as x approaches negative infinity of: (5x - 7x³)/(2x² + 14x³ - 9)
  • Both highest degrees are 3.
  • Result = -7 / 14 = -1/2.

Top-Heavy Rational Functions

• Example: (5x + 6x²)/(3x - 8)

  • Ignore the insignificant terms: limit = (6x²)/(3x) = 2x → positive infinity.

• Another Example: (5 + 2x - 3x³)/(4x² + 9x - 7)

  • Significant term is -3x³ in numerator and 4x² in denominator.
  • Result = (-3x³)/(4x²) = -3x/4 → positive infinity as x approaches negative infinity.