lim x 7 2 x 2 1 2 SOLUTION 1 Preliminary analysis of the problem ( guessing a value for ) Let be a given positive number We want to find a number such that If 0 x 7 then ( 2 x 2 ) 1 2 But ( 2 x 2 ) 1 2 2 x 1 4 2 Therefore, we want' such that If , 0 x 7 then , 2 1 that is , if 0 x 7 then r 2 This suggests that we should choose 2 2 Proof ( showing that works ) Given 0 , choose t 2 If 0 , then ( 2 x 2 ) 1 2 2

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Question: lim x 7 2 x - 2 = 1 2 SOLUTION 1 . Preliminary analysis of the problem ( guessing a value for ) .

lim_{x 7} 2 x - 2 = 12

SOLUTION

1 .

Preliminary analysis of the problem

(

guessing a value for

) .

Let

be a given positive number. We want to find a number

such that

0 |x-7|

then

| (2 x - 2) - 12 | .

But

| (2 x - 2) - 12 | - | 2 x - 14 | = 2

| .

Therefore, we want'

such that

, 0 |x-7|

then

, 2 | | 1

that is

,

0 |x-7|

then

\frac{r}{2}

This suggests that we should choose

= \frac{}{2} .

2 .

Proof

(

showing that

works

) .

Given

> 0,

choose

= \frac{t}{2} .

0,

then

| (2 x - 2) - 12 | = | |

= 2 | |

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