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Evaluate $space space space space space limit as straight x rightwards arrow 0 of fraction numerator square root of 1 plus straight x squared end root minus square root of 1 plus straight x end root over denominator square root of 1 plus straight x cubed end root space minus square root of 1 plus straight x end root end fraction$

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12.

Evaluate $space space space space space space space space space space space space limit as straight x rightwards arrow square root of 2 of fraction numerator straight x to the power of 9 minus 3 straight x to the power of 8 plus straight x to the power of 6 minus 4 straight x squared minus 16 straight x plus 84 over denominator straight x to the power of 5 minus 3 straight x to the power of 4 minus 4 straight x plus 12 end fraction$

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13.

Evaluate $space space space space space space space space limit as straight x rightwards arrow 1 of open parentheses fraction numerator 1 over denominator straight x minus 1 end fraction minus fraction numerator 3 left parenthesis straight x minus 2 right parenthesis over denominator straight x cubed minus 3 straight x squared plus 2 end fraction close parentheses$

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14.
Evaluate $space space space space space space limit as straight x rightwards arrow 0 of straight f left parenthesis straight x right parenthesis comma space where space straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left columnspacing 1.4ex end attributes row cell fraction numerator open vertical bar straight x close vertical bar over denominator straight x end fraction comma end cell cell straight x not equal to 0 end cell row cell 0 space comma end cell cell straight x equals 0 end cell end table close$

L.H.L = $space space space space space limit as straight x rightwards arrow 0 to the power of minus of space straight f left parenthesis straight x right parenthesis space equals space limit as straight x rightwards arrow 0 to the power of minus of fraction numerator open vertical bar straight x close vertical bar over denominator straight x end fraction$ [∵ here x < 0]

 $space space space space space space space space space space space equals space limit as straight x rightwards arrow 0 to the power of minus of straight f left parenthesis straight x right parenthesis space equals space limit as straight x rightwards arrow 0 to the power of minus of left parenthesis negative 1 right parenthesis equals negative 1 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space left square bracket because space for space straight x space less than 0 comma space open vertical bar x close vertical bar equals negative x right square bracket$

R.H.L. $space space space space equals space limit as straight x rightwards arrow 0 to the power of plus space of straight f left parenthesis straight x right parenthesis space equals limit as straight x rightwards arrow 0 to the power of plus of fraction numerator open curly brackets straight x close curly brackets over denominator straight x end fraction space space space space space space space space space space space space space space space space space space space space$ [∵ here x > 0]


 $space space space space space equals space limit as straight x rightwards arrow 0 to the power of plus of straight x over straight x$ [ ∵ for x >0, $space space space space space open vertical bar straight x close vertical bar equals straight x$ ]

$space space space space space space space space space space space space space equals space space limit as straight x rightwards arrow 0 to the power of plus of left parenthesis 1 right parenthesis equals 1$

Thus, L.H.L $space space not equal to$ R.H.L.

∴ $space space space limit as straight x rightwards arrow 0 of straight f left parenthesis straight x right parenthesis space$ does not exist.

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15.

Suppose that

$space space space space space space space space space space space straight f left parenthesis straight x right parenthesis space open curly brackets table row cell table row cell table row cell straight a plus bx comma space straight x space less than 1 end cell end table end cell end table end cell row cell table row cell table row cell 4 comma space straight x space equals space 1 end cell row cell straight b minus ax comma space straight x space greater than 1 end cell end table end cell end table end cell end table close$

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

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16.

Let f be a function defined by $space space space space straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left columnspacing 1.4ex end attributes row cell fraction numerator straight x over denominator open vertical bar straight x close vertical bar plus 2 straight x squared end fraction end cell cell straight x not equal to 0 end cell row cell 0 comma end cell cell straight x equals 0 end cell end table close$

Evaluate $space space limit as straight x rightwards arrow 0 to the power of asterisk times of straight f left parenthesis straight x right parenthesis comma$ what can you say about $space space space space space space space space space space space space limit as straight x rightwards arrow 0 of straight f left parenthesis straight x right parenthesis$ ?

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17.

Evaluate $space space space space space space limit as straight x rightwards arrow 0 of fraction numerator sin space ax over denominator sin space bs end fraction comma space straight a comma space straight b space not equal to 0.$

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18.

Evaluate $space space space space space limit as straight x rightwards arrow 0 of fraction numerator sin space ax over denominator bx end fraction.$

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19.

Evaluate $space space space space space space space limit as straight x rightwards arrow 0 of fraction numerator sin space 5 straight x over denominator straight x end fraction$

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20.

Evaluate $space space space space space space space space space space space space space space space space space limit as straight x rightwards arrow 0 of fraction numerator 1 minus cos space straight x over denominator straight x end fraction.$

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14. Evaluate