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

Since at $space straight x space equals space square root of 2 comma end root$ numerator and denominator becomes zero.

∴ it is of $0 over 0$ form.

∴ By factor theorem, $space space left parenthesis straight x minus square root of 2 right parenthesis$ is a factor of both numerator and denominator.

$rightwards double arrow$ $space space straight x space plus square root of 2$ is also a factor of both numerator and denominator because irrational roots occur in pairs.

So, x²-2 is a factor of both numerator and denominator.
By long division method, dividing numerator and denominator by x²-2.

x⁹- 3x⁸+ x⁶- 9x⁴- 4x²- 16x + 84 = (x²-2) (x⁷- 3x⁶+ 2x⁵-5x⁴+ 4x³-19x²+8x - 42) and x⁵-3x⁴-4x + 12 = (x²-2) (x³- 3x²+ 2x -6)

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

$space space space space space space equals space space space stack space lim with straight x rightwards arrow square root of 2 below fraction numerator left parenthesis straight x squared minus 2 right parenthesis left parenthesis straight x to the power of 7 minus 3 straight x to the power of 6 plus 2 straight x to the power of 5 minus 5 straight x to the power of 4 plus 4 straight x cubed minus 19 straight x squared plus 8 straight x minus 42 right parenthesis over denominator left parenthesis straight x squared minus 2 right parenthesis left parenthesis straight x cubed minus 3 straight x squared plus 2 straight x minus 6 right parenthesis end fraction$

$space space space space space space space space space equals space space fraction numerator left parenthesis square root of 2 right parenthesis end root to the power of 7 minus 3 left parenthesis square root of 2 right parenthesis to the power of 6 plus 2 left parenthesis square root of 2 right parenthesis end root to the power of 5 minus 5 left parenthesis square root of 2 right parenthesis to the power of 4 plus 4 left parenthesis square root of 2 right parenthesis end root cubed minus 19 left parenthesis square root of 2 right parenthesis end root squared plus 8 left parenthesis square root of 2 right parenthesis end root minus 42 over denominator left parenthesis square root of 2 right parenthesis cubed minus 3 left parenthesis square root of 2 right parenthesis squared plus 2 left parenthesis square root of 2 right parenthesis minus 6 end fraction$

$space space space space space space space space space equals space fraction numerator 8 square root of 2 minus 24 plus 8 square root of 2 minus 20 plus 8 square root of 2 minus 38 plus 8 square root of 2 minus 42 over denominator 2 square root of 2 minus 6 plus 2 square root of 2 minus 6 end fraction space space space space$ $space space space space open square brackets because space straight x space not equal to square root of 2 space rightwards double arrow space straight x squared minus 2 not equal to 0 close square brackets$

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

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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?

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

126 Views

19.

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

135 Views

20.

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

139 Views

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12. Evaluate