Limits at Infinity

Overview

Discuss how to find the limit of a function as x approaches infinity or negative infinity.

Function: x²
- If x becomes very large, x² becomes an extremely large value.
- Example: If x = 1000, then x² = 1,000,000.
- Conclusion: Limit as x approaches ∞ for x² is ∞.
Function: x³
- If x becomes very large, x³ becomes an extremely large positive number.
- If x becomes very large negative, x³ becomes an extremely large negative number.
- Conclusion: Limit as x approaches ∞ for x³ is ∞, and for -∞, it is -∞.

Function: x²
- If we replace x with -∞, squaring a negative number results in a positive number.
- Conclusion: Limit as x approaches -∞ for x² is ∞.
Function: x³
- If x approaches -∞ and is raised to the third power, the result is negative.
- Conclusion: Limit as x approaches -∞ for x³ is -∞.

Expression: 5 + 2x - x³
- As x approaches -∞, minor terms can be ignored.
- Conclusion: Limit is dominated by the term with the highest power, -x³.
- Final answer: Positive infinity (+∞).
Expression: 3x³ - 5x⁴
- As x approaches -∞, term -5x⁴ dominates.
- Negative times negative results in positive
- Final answer: Positive infinity (+∞).

Expression: 1/x
- As x approaches ∞, the value becomes smaller.
- Conclusion: Limit as x approaches ∞ for 1/x is 0.
Expression: (5x + 2) / (7x - x²)
- Degree of the denominator is higher than the numerator.
- Conclusion: Limit as x approaches ∞ for bottom-heavy functions is always 0.
Expression: (8x² - 5x) / (4x² + 7)
- Degree of numerator equals that of the denominator.
- Conclusion: Divide their coefficients, limit is 2.
Expression: (5x - 7x³)/(2x² + 14x³ - 9)
- Dominant degrees are the same:
- Result: -7 / 14 = -0.5
- Final answer: -0.5.
Expression: (5x + 6x²) / (3x - 8)
- Terms 5x and -8 are insignificant.
- Reduced form: 2x → 2 * ∞ = ∞.
- Conclusion: Final answer is positive infinity (∞).
Expression: (5 + 2x - 3x³) / (4x² + 9x - 7)
- Neglect terms with lower degrees
- Dominant term: -3x³ / 4x²
- Reduced to: -3/4 * x → -3 (∞) = ∞.