Limits and Derivatives

Limits and Derivatives

Question

Evaluate     <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>.

Answer

Evaluating the function at x =2, it is of the form space 0 over 0.
Therefor, space space space space space space space space space space space space space space space space space space space space space space space space limit as straight x rightwards arrow 2 of fraction numerator straight x cubed minus 4 straight x squared plus 4 straight x over denominator straight x squared minus 4 end fraction equals limit as straight x rightwards arrow 2 of fraction numerator straight x left parenthesis straight x squared minus 4 straight x plus 4 right parenthesis over denominator straight x squared minus 2 squared end fraction equals limit as straight x rightwards arrow 2 of fraction numerator straight x left parenthesis straight x minus 2 right parenthesis squared over denominator left parenthesis straight x minus 2 right parenthesis left parenthesis straight x plus 2 right parenthesis end fraction equals limit as straight x rightwards arrow 2 of fraction numerator straight x left parenthesis straight x minus 2 right parenthesis over denominator straight x plus 2 end fraction

                                                                                                                                                                                                                                                                                                                                                          space space space space space space space space space space space space space space space space space space space space space open parentheses because space straight x not equal to 2 rightwards double arrow negative 2 not equal to 0 close parentheses
          space space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator 3 left parenthesis 2 right parenthesis plus 5 over denominator 2 plus 2 end fraction equals 11 over 4

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