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

It is of 0 over 0 form.

∴         space 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 1 of fraction numerator left parenthesis straight x plus straight x squared plus straight x cubed plus.... plus straight x to the power of straight n right parenthesis minus straight n over denominator straight x minus 1 end fraction equals limit as straight x rightwards arrow 1 of fraction numerator left parenthesis straight x minus 1 right parenthesis plus left parenthesis straight x squared minus 1 right parenthesis plus left parenthesis straight x cubed minus 1 right parenthesis plus... plus left parenthesis straight x to the power of straight n minus 1 right parenthesis over denominator straight x minus 1 end fraction

    space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space limit as x rightwards arrow 1 of open square brackets 1 plus fraction numerator x squared minus 1 over denominator x minus 1 end fraction plus fraction numerator x cubed minus 1 over denominator x minus 1 end fraction plus.... plus fraction numerator x to the power of n minus 1 over denominator x minus 1 end fraction close square brackets
                  space space space space space space space space space space space space space space space space space space space space space equals space limit as straight x rightwards arrow 1 of 1 plus limit as straight x rightwards arrow of fraction numerator straight x squared minus 1 over denominator straight x minus 1 end fraction plus limit as straight x rightwards arrow 1 of fraction numerator straight x cubed minus 1 over denominator straight x minus 1 end fraction plus...... plus limit as straight x rightwards arrow 1 of fraction numerator straight x to the power of straight n minus 1 over denominator straight x minus 1 end fraction
                                   = 1 + 2(1)+ 3(1)+ ......+ n(1)n-1 = 1 + 2 + 3 + ..... +n
                                      
                            space space space space space space space space space equals space sum subscript straight n fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction
 


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