1. 1 As x approaches a finite value f(x) can approach a finite value f(x) can become arbitrarily large As x becomes arbitrarily large Limits

2. 2 Some limits do not exist Worked example Show that As x approaches a finite value f(x) can approach a finite value f(x) can become arbitrarily large As x becomes arbitrarily large Limits

3. 3 x Notation Symbolic meaning Vernacular As x becomes arbitrarily close to a, f(x) becomes arbitrarily close to L. y

4. y 4 xNotation Symbolic meaning Vernacular As x becomes arbitrarily close to a, f(x) becomes arbitrarily close to L. a L

5. y a L 5 xNotation Symbolic meaning Vernacular As x becomes arbitrarily close to a, f(x) becomes arbitrarily close to L.

6. y a L 6 xNotation Symbolic meaning Vernacular As x becomes arbitrarily close to a, f(x) becomes arbitrarily close to L.

7. y a L 7 x STOP Can L still be the limit of f(x) as x approaches a if, as illustrated, f(a) no longer equals L? Notation Vernacular As x becomes arbitrarily close to a, f(x) becomes arbitrarily close to L.

8. 8 Some limits do not exist Worked example Show that As x approaches a finite value f(x) can approach a finite value f(x) can become arbitrarily large As x becomes arbitrarily large Limits

9. 9 Definition of improper limit of a functionNotation Symbolic meaning Vernacular As x becomes arbitrarily close to a, f(x) becomes arbitrarily large. x y

10. M 10 Definition of improper limit of a functionNotation Symbolic meaning Vernacular As x becomes arbitrarily close to a, f(x) becomes arbitrarily large. a x y

11. M 11 Definition of improper limit of a functionNotation Symbolic meaning Vernacular As x becomes arbitrarily close to a, f(x) becomes arbitrarily large. a x y

12. M 12 Definition of improper limit of a function Notation Symbolic meaning Vernacular As x becomes arbitrarily close to a, f(x) becomes arbitrarily large. a x y

13. M 13 Definition of improper limit of a functionNotation Symbolic meaning Vernacular As x becomes arbitrarily close to a, f(x) becomes arbitrarily large. a x y STOP

14. 14 Some limits do not exist Worked example Show that As x approaches a finite value f(x) can approach a finite value f(x) can become arbitrarily large As x becomes arbitrarily large Limits

15. y 15 x Notation Symbolic meaning Vernacular As x becomes arbitrarily large, f(x) becomes arbitrarily close to L.

16. x y S 16 L Notation Symbolic meaning Vernacular As x becomes arbitrarily large, f(x) becomes arbitrarily close to L.

17. x y S 17 L Notation Symbolic meaning Vernacular As x becomes arbitrarily large, f(x) becomes arbitrarily close to L.

18. x y S 18 LNotation Symbolic meaning Vernacular As x becomes arbitrarily large, f(x) becomes arbitrarily close to L.

19. 19 Some limits do not exist Worked example Show that As x approaches a finite value f(x) can approach a finite value f(x) can become arbitrarily large As x becomes arbitrarily large Limits

20. y 20 Improper limits at infinity x Notation Symbolic meaning Vernacular As x becomes arbitrarily large, f(x) becomes arbitrarily large.

21. M x y S 21 Improper limits at infinityNotation Symbolic meaning Vernacular As x becomes arbitrarily large, f(x) becomes arbitrarily large.

22. M x y S 22 Improper limits at infinityNotation Symbolic meaning Vernacular As x becomes arbitrarily large, f(x) becomes arbitrarily large.

23. M x y S 23 Improper limits at infinity Notation Symbolic meaning Vernacular As x becomes arbitrarily large, f(x) becomes arbitrarily large.

24. Worked example Show that 24 Some limits do not exist As x approaches a finite value f(x) can approach a finite value f(x) can become arbitrarily large As x becomes arbitrarily large Limits

25. yy 25 Sometimes a limit does not exist (D.N.E.) Example: Limit as x becomes arbitrarily large x L Example: Limit of a function a L x S

26. Worked example Show that 26 Some limits do not exist As x approaches a finite value f(x) can approach a finite value f(x) can become arbitrarily large As x becomes arbitrarily large Limits

27. y 2 27 Show that 1 x Want to show

28. y 2 28 1 x y MAX y MIN Show that Want to show

29. 29 Some limits do not exist Worked example Show that As x approaches a finite value f(x) can approach a finite value f(x) can become arbitrarily large As x becomes arbitrarily large Limits

30. a M x y 30 Definition of improper limit of a function Notation Symbolic meaning Vernacular As x becomes arbitrarily close to a, f(x) becomes arbitrarily large.