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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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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

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

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

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

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

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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.

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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

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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

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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

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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

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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

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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

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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

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

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

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

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

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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

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

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

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

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

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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

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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

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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

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y 2 27 Show that 1 x Want to show

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

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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

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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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§2 Limits and continuity 2.1 Some important definitions and results about functions.

§2 Limits and continuity 2.1 Some important definitions and results about functions.

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