Limits

1. Limits y y y Limit of a function Improper limits Limits “at” infinity Show that y Worked example Sometimes limits do not exist x x x x

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

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

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

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

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

7. Limits y y y Limit of a function Improper limits Limits “at” infinity Show that y Worked example Sometimes limits do not exist x x x x

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

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

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

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

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

13. Limits y y y Limit of a function Improper limits Limits “at” infinity Show that y Worked example Sometimes limits do not exist x x x x

14. 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 or below L. x

15. 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 or below L. x S

16. 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 or 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 or below L. x S

18. Limits y y y Limit of a function Improper limits Limits “at” infinity Show that y Worked example Sometimes limits do not exist x x x x

19. 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 p+ p- S p

20. Limits y y y Limit of a function Improper limits Limits “at” infinity Show that y Worked example Sometimes limits do not exist x x x x

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

22. 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