Limits & Continuity

Limits & ContinuityA limit describes the behaviour of functions.

Limits & ContinuityA limit describes the behaviour of functions.

( )lim :x af x

→

Limits & ContinuityA limit describes the behaviour of functions.

( )lim :x af x

→as the x value approaches a, what value does f(x) approach?

Limits & ContinuityA limit describes the behaviour of functions.

( )lim :x af x

→as the x value approaches a, what value does f(x) approach?

( )lim :x a

f x−→

Limits & ContinuityA limit describes the behaviour of functions.

( )lim :x af x

→as the x value approaches a, what value does f(x) approach?

( )lim :x a

f x−→

as the x value approaches a from the negative side, what value does f(x) approach?

Limits & ContinuityA limit describes the behaviour of functions.

( )lim :x af x

→as the x value approaches a, what value does f(x) approach?

( )lim :x a

f x−→

as the x value approaches a from the negative side, what value does f(x) approach?

y

x1

1

1y x= −

Limits & ContinuityA limit describes the behaviour of functions.

( )lim :x af x

→as the x value approaches a, what value does f(x) approach?

( )lim :x a

f x−→

as the x value approaches a from the negative side, what value does f(x) approach?

y

x1

1

1y x= −

1lim 1 0x

x−→

− =

Limits & ContinuityA limit describes the behaviour of functions.

( )lim :x af x

→as the x value approaches a, what value does f(x) approach?

( )lim :x a

f x−→

as the x value approaches a from the negative side, what value does f(x) approach?

y

x1

1

1y x= −

1lim 1 0x

x−→

− =

y

x

46

( )y f x=

Limits & ContinuityA limit describes the behaviour of functions.

( )lim :x af x

→as the x value approaches a, what value does f(x) approach?

( )lim :x a

f x−→

as the x value approaches a from the negative side, what value does f(x) approach?

y

x1

1

1y x= −

1lim 1 0x

x−→

− =

y

x

46

( )y f x=

( )0

lim 4x

f x−→

=

Limits & ContinuityA limit describes the behaviour of functions.

( )lim :x af x

→as the x value approaches a, what value does f(x) approach?

( )lim :x a

f x−→

as the x value approaches a from the negative side, what value does f(x) approach?

y

x1

1

1y x= −

1lim 1 0x

x−→

− =

y

x

46

( )y f x=

( )0

lim 4x

f x−→

=

( )lim :x a

f x+→

Limits & ContinuityA limit describes the behaviour of functions.

( )lim :x af x

→as the x value approaches a, what value does f(x) approach?

( )lim :x a

f x−→

as the x value approaches a from the negative side, what value does f(x) approach?

y

x1

1

1y x= −

1lim 1 0x

x−→

− =

y

x

46

( )y f x=

( )0

lim 4x

f x−→

=

( )lim :x a

f x+→

as the x value approaches a from the positive side, what value does f(x) approach?

Limits & ContinuityA limit describes the behaviour of functions.

( )lim :x af x

→as the x value approaches a, what value does f(x) approach?

( )lim :x a

f x−→

as the x value approaches a from the negative side, what value does f(x) approach?

y

x1

1

1y x= −

1lim 1 0x

x−→

− =

y

x

46

( )y f x=

( )0

lim 4x

f x−→

=

( )lim :x a

f x+→

as the x value approaches a from the positive side, what value does f(x) approach?

1lim 1 0x

x+→

− =

Limits & ContinuityA limit describes the behaviour of functions.

( )lim :x af x

→as the x value approaches a, what value does f(x) approach?

( )lim :x a

f x−→

as the x value approaches a from the negative side, what value does f(x) approach?

y

x1

1

1y x= −

1lim 1 0x

x−→

− =

y

x

46

( )y f x=

( )0

lim 4x

f x−→

=

( )lim :x a

f x+→

as the x value approaches a from the positive side, what value does f(x) approach?

1lim 1 0x

x+→

− = ( )0

lim 6x

f x+→

=

Limits & ContinuityA limit describes the behaviour of functions.

( )lim :x af x

→as the x value approaches a, what value does f(x) approach?

( )lim :x a

f x−→

as the x value approaches a from the negative side, what value does f(x) approach?

y

x1

1

1y x= −

1lim 1 0x

x−→

− =

y

x

46

( )y f x=

( )0

lim 4x

f x−→

=

( )lim :x a

f x+→

as the x value approaches a from the positive side, what value does f(x) approach?

1lim 1 0x

x+→

− = ( )0

lim 6x

f x+→

=

( ) ( ) ( )If lim lim , then is continuous at x a x a

f x f x f x x a− +→ →

= =

Finding Limits

Finding Limits(1) Direct Substitution

Finding Limits(1) Direct Substitution

5e.g. lim 7

xx

→+

Finding Limits(1) Direct Substitution

5e.g. lim 7

xx

→+ 5 7

12

= +=

Finding Limits(1) Direct Substitution

5e.g. lim 7

xx

→+ 5 7

12

= +=

(2) Factorise and Cancel

Finding Limits(1) Direct Substitution

5e.g. lim 7

xx

→+ 5 7

12

= +=

(2) Factorise and Cancel2

3

9e.g. lim

3x

xx→

−−

Finding Limits(1) Direct Substitution

5e.g. lim 7

xx

→+ 5 7

12

= +=

(2) Factorise and Cancel2

3

9e.g. lim

3x

xx→

−−

( ) ( )( )3

3 3lim

3x

x x

x→

+ −=

−

Finding Limits(1) Direct Substitution

5e.g. lim 7

xx

→+ 5 7

12

= +=

(2) Factorise and Cancel2

3

9e.g. lim

3x

xx→

−−

( ) ( )( )3

3 3lim

3x

x x

x→

+ −=

−( )

3lim 3x

x→

= +

Finding Limits(1) Direct Substitution

5e.g. lim 7

xx

→+ 5 7

12

= +=

(2) Factorise and Cancel2

3

9e.g. lim

3x

xx→

−−

( ) ( )( )3

3 3lim

3x

x x

x→

+ −=

−( )

3lim 3x

x→

= +

3 3

6

= +=

(3) Special Limit

(3) Special Limit1

lim 0x x→∞

=

(3) Special Limit1

lim 0x x→∞

=

( )3 2

3

3 2 1e.g. lim

4 1x

x x xi

x→∞

+ + −−

(3) Special Limit1

lim 0x x→∞

=

( )3 2

3

3 2 1e.g. lim

4 1x

x x xi

x→∞

+ + −−

3 2

3 3 3 3

3

3 3

3 2 1

lim4 1x

x x xx x x x

xx x

→∞

+ + −=

−

(3) Special Limit1

lim 0x x→∞

=

( )3 2

3

3 2 1e.g. lim

4 1x

x x xi

x→∞

+ + −−

3 2

3 3 3 3

3

3 3

3 2 1

lim4 1x

x x xx x x x

xx x

→∞

+ + −=

−

2 3

3

3 2 11

lim1

4x

x x x

x

→∞

+ + −=

−

(3) Special Limit1

lim 0x x→∞

=

( )3 2

3

3 2 1e.g. lim

4 1x

x x xi

x→∞

+ + −−

3 2

3 3 3 3

3

3 3

3 2 1

lim4 1x

x x xx x x x

xx x

→∞

+ + −=

−

2 3

3

3 2 11

lim1

4x

x x x

x

→∞

+ + −=

−14

=

(3) Special Limit1

lim 0x x→∞

=

( )3 2

3

3 2 1e.g. lim

4 1x

x x xi

x→∞

+ + −−

3 2

3 3 3 3

3

3 3

3 2 1

lim4 1x

x x xx x x x

xx x

→∞

+ + −=

−

2 3

3

3 2 11

lim1

4x

x x x

x

→∞

+ + −=

−14

=

( )2

3

4 lim

1x

x xii

x→∞

−+

(3) Special Limit1

lim 0x x→∞

=

( )3 2

3

3 2 1e.g. lim

4 1x

x x xi

x→∞

+ + −−

3 2

3 3 3 3

3

3 3

3 2 1

lim4 1x

x x xx x x x

xx x

→∞

+ + −=

−

2 3

3

3 2 11

lim1

4x

x x x

x

→∞

+ + −=

−14

=

( )2

3

4 lim

1x

x xii

x→∞

−+

010

=

=

(3) Special Limit1

lim 0x x→∞

=

( )3 2

3

3 2 1e.g. lim

4 1x

x x xi

x→∞

+ + −−

3 2

3 3 3 3

3

3 3

3 2 1

lim4 1x

x x xx x x x

xx x

→∞

+ + −=

−

2 3

3

3 2 11

lim1

4x

x x x

x

→∞

+ + −=

−14

=

( )2

3

4 lim

1x

x xii

x→∞

−+

010

=

=

( )7 6 2

7 lim3 974x

x x xiii

x x→∞

+ +− −

(3) Special Limit1

lim 0x x→∞

=

( )3 2

3

3 2 1e.g. lim

4 1x

x x xi

x→∞

+ + −−

3 2

3 3 3 3

3

3 3

3 2 1

lim4 1x

x x xx x x x

xx x

→∞

+ + −=

−

2 3

3

3 2 11

lim1

4x

x x x

x

→∞

+ + −=

−14

=

( )2

3

4 lim

1x

x xii

x→∞

−+

010

=

=

( )7 6 2

7 lim3 974x

x x xiii

x x→∞

+ +− −

13

=

( )3

2

2 lim

1x

xiv

x→∞

+−

( )3

2

2 lim

1x

xiv

x→∞

+−

10

=

= ∞

( )3

2

2 lim

1x

xiv

x→∞

+−

10

=

= ∞

( ) ( ) ( )( ) ( )

3 2 Find the horizontal asymptote of

1 1

x xv y

x x

+ −=

− +

( )3

2

2 lim

1x

xiv

x→∞

+−

10

=

= ∞

( ) ( ) ( )( ) ( )

3 2 Find the horizontal asymptote of

1 1

x xv y

x x

+ −=

− +( ) ( )( ) ( )

3 2lim

1 1x

x x

x x→∞

+ −− +

( )3

2

2 lim

1x

xiv

x→∞

+−

10

=

= ∞

( ) ( ) ( )( ) ( )

3 2 Find the horizontal asymptote of

1 1

x xv y

x x

+ −=

− +( ) ( )( ) ( )

3 2lim

1 1x

x x

x x→∞

+ −− +

2

2

6lim

1x

x xx→∞

+ −=−

( )3

2

2 lim

1x

xiv

x→∞

+−

10

=

= ∞

( ) ( ) ( )( ) ( )

3 2 Find the horizontal asymptote of

1 1

x xv y

x x

+ −=

− +( ) ( )( ) ( )

3 2lim

1 1x

x x

x x→∞

+ −− +

2

2

6lim

1x

x xx→∞

+ −=−

111

=

=

( )3

2

2 lim

1x

xiv

x→∞

+−

10

=

= ∞

( ) ( ) ( )( ) ( )

3 2 Find the horizontal asymptote of

1 1

x xv y

x x

+ −=

− +( ) ( )( ) ( )

3 2lim

1 1x

x x

x x→∞

+ −− +

2

2

6lim

1x

x xx→∞

+ −=−

111

=

=horizontal asymptote is 1y∴ =

( )3

2

2 lim

1x

xiv

x→∞

+−

10

=

= ∞

( ) ( ) ( )( ) ( )

3 2 Find the horizontal asymptote of

1 1

x xv y

x x

+ −=

− +( ) ( )( ) ( )

3 2lim

1 1x

x x

x x→∞

+ −− +

2

2

6lim

1x

x xx→∞

+ −=−

111

=

=horizontal asymptote is 1y∴ = Exercise 7I; 1a, 2ace, 3ac,

4a, 5ad, 8a, 9ab, 10a