Limits and Limits at infinity

1. Limits and Limits at infinity Limit Review 1

2. Limits can be calculated 3 ways • Numerically • Graphically • Analytically (direct substitution) • Properties of limits (pages 56-58)

3. Graphically • Determine the value of y as x approaches some number c

4. Formal Definition of a Limit

5. Analytic techniques • Direct substitution • Factoring- cancel common factors • Rationalize- multiply by the conjugate

6. Find 2) Find

7. 3) Find

8. One sided limits • left • Right • Limit existence

9. 4) Find

10. 5) Find

11. Infinite Limits • Limit of f(x) increases or decreases without bound as x approaches some number c • Limit does not exist DNE • Vertical asymptotes occur when limits are infinite • Limit as x approaches 1

12. 6) Find

13. 7) Find

14. 8) Find

15. 9) Find

16. Limits at Infinity • Identifies the behavior of the function as x tends to infinity • If , then L is a horizontal asymptote • Usually f(x) is rational • Limit as x approaches infinity

17. 10) Find

18. 11) Find

19. Limits at Infinity Shortcuts • If the degree is higher in the denominator, then the limit is equal to zero • If the degree is higher in the numerator then the limit does not exist • If the degrees are the same, then the limit equals the ratio of the coefficients BOBO BOTN EATS DC

20. 12) Find 13) Find

21. 14) Find and all asymptotes

22. 15) Find and all asymptotes

23. Home Work Limits Practice worksheet