As x approaches a finite value

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

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

3. definition of limit of a function y Notation Vernacular As x becomes arbitrarily close toa,f(x)becomes arbitrarily close toL. Symbolic meaning English meaning For all positive , there exists a positive such that confining x to between a – and a and between a and a + , implies that f(x) is also confined to a vertical range between above and below y = L. x

4. definition of limit of a function y Notation L+ Vernacular As x becomes arbitrarily close toa,f(x)becomes arbitrarily close toL. L Symbolic meaning English meaning For all positive , there exists a positive such that confining x to between a – and a and between a and a + , implies that f(x) is also confined to a vertical range between above and below y = L. L- x a a+ a-

5. definition of limit of a function y Notation Vernacular As x becomes arbitrarily close toa,f(x)becomes arbitrarily close toL. L+ L Symbolic meaning L- English meaning For all positive , there exists a positive such that confining x to between a – and a and between a and a + , implies that f(x) is also confined to a vertical range between above and below y = L. x a a+ a-

6. definition of limit of a function y Notation Vernacular As x becomes arbitrarily close toa,f(x)becomes arbitrarily close toL. L+ L Symbolic meaning L- English meaning For all positive , there exists a positive such that confining x to between a – and a and between a and a + , implies that f(x) is also confined to a vertical range between above and below y = L. x a a+ a-

7. definition of limit of a function y Notation Vernacular As x becomes arbitrarily close toa,f(x)becomes arbitrarily close toL. L+ Can L still be the limit of f(x) as x approaches a if, as illustrated, f(a) no longer equals L? L L- English meaning For all positive , there exists a positive such that confining x to between a – and a and between a and a + , implies that f(x) is also confined to a vertical range between above and below y = L. STOP x a a+ a-

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

9. Definition of improper limit of a function y Notation Vernacular As x becomes arbitrarily close to a, f(x) becomes arbitrarily large. Symbolic meaning English meaning For all positive , there exists a positive such that confining x to between a– and aand between aand a+ , implies that f(x) exceeds . x

10. Definition of improper limit of a function y Notation Vernacular As x becomes arbitrarily close to a, f(x) becomes arbitrarily large. M Symbolic meaning English meaning For all positive , there exists a positive such that confining x to between a– and aand between aand a+ , implies that f(x) exceeds . x a a+ a-

11. Definition of improper limit of a function y Notation Vernacular As x becomes arbitrarily close to a, f(x) becomes arbitrarily large. M Symbolic meaning English meaning For all positive , there exists a positive such that confining x to between a– and aand between aand a+ , implies that f(x) exceeds . x a a+ a-

12. Definition of improper limit of a function y Notation Vernacular As x becomes arbitrarily close to a, f(x) becomes arbitrarily large. M Symbolic meaning English meaning For all positive , there exists a positive such that confining x to between a– and aand between aand a+ , implies that f(x) exceeds . x a a+ a-

13. Definition of improper limit of a function y Notation Vernacular As x becomes arbitrarily close to a, f(x) becomes arbitrarily large. M Symbolic meaning English meaning For all positive , there exists a positive such that confining x to between a– and aand between aand a+ , implies that f(x) exceeds . Write down the analogous definition for STOP x a a+ a-

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

15. Definition of limit of a function as y Notation Vernacular As x becomes arbitrarily large, f(x) becomes arbitrarily close to L. Symbolic meaning English meaning For all positive , there exists a positive such that insisting that x exceed implies that f(x) is between above and below L. x

16. Definition of limit of a function as y Notation L+ Vernacular As x becomes arbitrarily large, f(x) becomes arbitrarily close to L. L Symbolic meaning L- English meaning For all positive , there exists a positive such that insisting that x exceed implies that f(x) is between above and below L. x S

17. Definition of limit of a function as y Notation Vernacular As x becomes arbitrarily large, f(x) becomes arbitrarily close to L. L+ L L- Symbolic meaning English meaning For all positive , there exists a positive such that insisting that x exceed implies that f(x) is between above and below L. x S

18. Definition of limit of a function as y Notation Vernacular As x becomes arbitrarily large, f(x) becomes arbitrarily close to L. L+ L L- Symbolic meaning English meaning For all positive , there exists a positive such that insisting that x exceed implies that f(x) is between above and below L. x S

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

20. Improper limits at infinity y Notation Vernacular As x becomes arbitrarily large, f(x) becomes arbitrarily large. Symbolic meaning English meaning For all positive , there exists a positive such that insisting that x exceed implies that f(x) exceeds . x

21. Improper limits at infinity y Notation Vernacular As x becomes arbitrarily large, f(x) becomes arbitrarily large. M Symbolic meaning English meaning For all positive , there exists a positive such that insisting that x exceed implies that f(x) exceeds . x S

22. Improper limits at infinity y Notation Vernacular As x becomes arbitrarily large, f(x) becomes arbitrarily large. M Symbolic meaning English meaning For all positive , there exists a positive such that insisting that x exceed implies that f(x) exceeds . x S

23. Improper limits at infinity y Notation Vernacular As x becomes arbitrarily large, f(x) becomes arbitrarily large. M Symbolic meaning English meaning For all positive , there exists a positive such that insisting that x exceed implies that f(x) exceeds . x S

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

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

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

27. Example: Want to show Show that y 2+ 2 2- x 1+ 1 1-

28. Example: Want to show Works for any Show that Pick an arbitrary y 2+ Pick an . Can we choose s.t.? 2 yMAX 2- yMIN x 1+ 1 1-

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