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A problem on epsilon and delta in the proof of L'Hospital's Rule

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1

1

$begingroup$

I'm following the proof of L'Hospital's Rule from a text but have trouble understanding this part regarding epsilon-delta.

Assume $Lgt 0$ and $epsilon gt 0$ is given and $delta in (0,1)$. What we have is

$$(L-epsilon)(1-delta)-deltalt fracf(alpha)g(alpha)lt(L+epsilon)+delta$$

If we take $delta=operatornamemin1,epsilon,epsilon/($ then
$$L-2epsilonle fracf(alpha)g(alpha)le L+2epsilon$$

I can't get the first inequality above. Also, shouldn't we get a strict inequality on the right side? I've tried some manipulations but all have been unsuccessful in getting that. How can I show this part?

real-analysis analysis

share|cite|improve this question

edited Mar 14 at 7:13

Winfield Chen

484

asked Mar 13 '15 at 0:23

takecaretakecare

2,34021537

$endgroup$

add a comment | 

1

1

$begingroup$

I'm following the proof of L'Hospital's Rule from a text but have trouble understanding this part regarding epsilon-delta.

Assume $Lgt 0$ and $epsilon gt 0$ is given and $delta in (0,1)$. What we have is

$$(L-epsilon)(1-delta)-deltalt fracf(alpha)g(alpha)lt(L+epsilon)+delta$$

If we take $delta=operatornamemin1,epsilon,epsilon/($ then
$$L-2epsilonle fracf(alpha)g(alpha)le L+2epsilon$$

I can't get the first inequality above. Also, shouldn't we get a strict inequality on the right side? I've tried some manipulations but all have been unsuccessful in getting that. How can I show this part?

real-analysis analysis

share|cite|improve this question

edited Mar 14 at 7:13

Winfield Chen

484

asked Mar 13 '15 at 0:23

takecaretakecare

2,34021537

$endgroup$

add a comment | 

$begingroup$

I'm following the proof of L'Hospital's Rule from a text but have trouble understanding this part regarding epsilon-delta.

Assume $Lgt 0$ and $epsilon gt 0$ is given and $delta in (0,1)$. What we have is

$$(L-epsilon)(1-delta)-deltalt fracf(alpha)g(alpha)lt(L+epsilon)+delta$$

If we take $delta=operatornamemin1,epsilon,epsilon/($ then
$$L-2epsilonle fracf(alpha)g(alpha)le L+2epsilon$$

I can't get the first inequality above. Also, shouldn't we get a strict inequality on the right side? I've tried some manipulations but all have been unsuccessful in getting that. How can I show this part?

real-analysis analysis

share|cite|improve this question

edited Mar 14 at 7:13

Winfield Chen

484

asked Mar 13 '15 at 0:23

takecaretakecare

2,34021537

$endgroup$

share|cite|improve this question

edited Mar 14 at 7:13

Winfield Chen

484

asked Mar 13 '15 at 0:23

takecaretakecare

2,34021537

share|cite|improve this question

edited Mar 14 at 7:13

Winfield Chen

484

asked Mar 13 '15 at 0:23

takecaretakecare

2,34021537

edited Mar 14 at 7:13

Winfield Chen

484

edited Mar 14 at 7:13

Winfield Chen

484

asked Mar 13 '15 at 0:23

takecaretakecare

2,34021537

asked Mar 13 '15 at 0:23

takecaretakecare

2,34021537