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

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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?

L.H.L. = $space space space space space space space space space limit as straight x rightwards arrow 1 to the power of minus of straight f left parenthesis straight x right parenthesis space equals limit as straight x rightwards arrow 1 to the power of minus of left parenthesis straight a plus bx right parenthesis$ [∵ here x <1]

Put x = 1-4, h>0 and h $rightwards arrow$ 0

∴ L.H.L = $space space space space space space space space space space space stack space lim with h rightwards arrow 0 below left square bracket a plus b left parenthesis 1 minus h right parenthesis right square bracket space equals space a plus b left parenthesis 1 minus 0 right parenthesis equals a plus b$

R.H.L. = $space space space space limit as straight x rightwards arrow 1 to the power of plus of straight f left parenthesis straight x right parenthesis space equals space limit as straight x rightwards arrow 1 to the power of plus of left parenthesis straight b minus ax right parenthesis$ [∵ here x > 1]

Put x = 1-h, h>0 and h $rightwards arrow$ 0

∴ R.H.L = $space space space space space space space limit as straight h rightwards arrow 0 of left square bracket straight b minus straight a left parenthesis 1 plus straight h right parenthesis right square bracket equals straight b minus straight a left parenthesis 1 plus 0 right parenthesis equals straight b minus straight a$

Also, f(1) = 4

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

$rightwards double arrow$
$space space space space space space limit as straight x rightwards arrow 1 to the power of minus of straight f left parenthesis straight x right parenthesis space equals straight f left parenthesis 1 right parenthesis space and space limit as straight x rightwards arrow 1 to the power of plus of straight f left parenthesis straight x right parenthesis space equals space straight f left parenthesis 1 right parenthesis$

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

Adding these equations, we get 2b = 8 $rightwards double arrow$ b = 4

∴ a = 4 - b = 4 - 4 = 0

Hence, a = 0, b = 4

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