well it was working and stopped suddenly.this are the screen shot for the database is attached.

Well, I'd say your problem doesn't come from LimeSurvey, but rather from a kind of Database corruption...

example:
stackoverflow.com/questions/7759170/mysq...it-does-or-it-should

once you fixed it, you should still update.

​i run mysqlcheck over the table, and it says ok plus when i execute show tables over the db it shows the list of the tables properly and also select * from lime_settings_global it displays the content. And also i have two limeservay databases and i tried to connect on both dbs by changing the dbname from application/config/config.php and same error is shown. I have another system on my server using mysql and all are working fine.​
So do u think the problem is with the db?

what else can it be the problem ​

well the main point would be to know what changed those last days:

- update of the server: PHP/MySQL/Apache?
- installed/activated plugins in limeSurvey, apache, etc?
- changes in global settings?

Also, what version of PHP are you using? PHP 5.3 makes a lot of weird unpredictable problems (most buggy PHP version ever).

i have two limeservay databases and i tried to connect on both dbs by changing the dbname from application/config/config.php and same error is shown

So now, just try the contrary: backup everything, install a fresh up to date LimeSurvey (2.06LTS or 2.5), connect it to the old DB, and see if it solves the problem.