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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.

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Presentation on theme: "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."— Presentation transcript:

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 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.


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