Given: 2(3x+4)=20, prove x=2
In algebraic proof

(

4

+

x

−

3

x

3

)

−

2

=

−

(

3

x

3

−

x

−

2

)

=

−

(

x

−

1

)

(

3

x

2

+

3

x

+

2

)

So

∣

∣

(

4

+

x

−

3

x

3

)

−

2

∣

∣

=

|

x

−

1

|

⋅

∣

∣

3

x

2

+

3

x

+

2

∣

∣

Given

ε

>

0

, let

δ

=

min

{

1

,

ε

20

}

and note that

δ

>

0

If

0

<

|

x

−

1

|

<

1

then

0

<

x

<

2

so that

2

<

(

3

x

2

+

3

x

+

2

)

<

20

and

∣

∣

3

x

2

+

3

x

+

2

∣

∣

<

20

So,

∣

∣

(

4

+

x

−

3

x

3

)

−

2

∣

∣

=

|

x

−

1

|

⋅

∣

∣

3

x

2

+

3

x

+

2

∣

∣

<

|

x

−

1

|

⋅

20

<

ε

20

⋅

20

=

ε

We have shown that for any positive

ε

, there is a positive

δ

such that

If

0

<

|

x

−

1

|

<

1

, then

∣

∣

(

4

+

x

−

3

x

3

)

−

2

∣

∣

<

ε

So, by the definition of limit,  

lim

x

→

1

(

4

+

x

−

3

x

2

)

=

2

Step-by-step explanation:

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