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Limits and Derivatives

Short Answer Type

81.

Evaluate the following limits :

$space space space space space space space space space space space space space space space space space space left parenthesis straight i right parenthesis space space limit as straight x rightwards arrow 1 of fraction numerator straight x to the power of 15 minus 1 over denominator straight x to the power of 10 minus 1 end fraction space space space space space space space space space space space space space space space space space space space space space space space space space left parenthesis ii right parenthesis space limit as straight x rightwards arrow 0 of fraction numerator left parenthesis straight x plus 1 right parenthesis to the power of 5 minus 1 over denominator straight x end fraction$

121 Views

82.

Find a, so that $space space space space space space space space space limit as straight x rightwards arrow negative 1 of straight f left parenthesis straight x right parenthesis space$ may exist, where $space space space space space space space space space space straight f left parenthesis straight x right parenthesis space equals space open curly brackets table attributes columnalign left end attributes row cell straight x squared plus 5 comma space straight x less or equal than negative 1 end cell row cell 3 straight x plus straight a comma space straight x space greater than space minus 1 end cell end table close$

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

Does $space space space space limit as straight z rightwards arrow 1 of straight f left parenthesis straight x right parenthesis$ exist? If no, explain.

103 Views

83.

Find the left limit of the function f defined by $space space straight f left parenthesis straight x right parenthesis space equals space square root of 6 minus 3 straight x end root space as space straight x rightwards arrow 2.$ Does the right limit exis as $space space straight x rightwards arrow 2 ?$

95 Views

84.

Evaluate $space space space limit as straight x rightwards arrow straight a of fraction numerator straight x cubed minus straight a cubed over denominator straight x squared minus straight a squared end fraction semicolon space space straight a not equal to 0.$

119 Views

85.

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

101 Views

86.
Evaluate $space space space space space space space space space space space space space space space space space space space space space space left parenthesis straight i right parenthesis space stack space lim with straight x rightwards arrow 0 below fraction numerator tan space straight x space minus space sin space straight x over denominator sin cubed straight x end fraction space space space space space space space space space space space space space left parenthesis ii right parenthesis space space limit as straight x rightwards arrow 0 space of fraction numerator sinx minus 2 sin 3 straight x minus sin 5 straight x over denominator straight x end fraction$

space space space space space space space space left parenthesis straight i right parenthesis space 1 half

(ii) 0

97 Views

87.

Evaluate $space space space space space space space space space space space limit as straight x rightwards arrow 0 of fraction numerator 2 sin space straight x space minus sin space 2 straight x over denominator straight x cubed end fraction$

111 Views

88.

Evaluate : $space space space space space space space space space limit as straight x rightwards arrow straight n divided by 2 of left parenthesis sec space straight x space minus space tan space straight x space right parenthesis$

96 Views

89.

Evaluate : $space space space space space space space space space space limit as straight X rightwards arrow 0 equals straight N divided by 6 of fraction numerator square root of 3 sin space straight x space minus space cos space straight x over denominator straight x space minus space straight pi divided by 6 end fraction$

127 Views

90.

Evaluate : $space space space space space space space space space space space space limit as straight x rightwards arrow straight pi divided by 6 of fraction numerator cot squared straight x minus 3 over denominator cosec space straight x space minus 2 end fraction$

100 Views

Previous Year Papers

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Short Answer Type

86. Evaluate