Limits and Derivatives

Question

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

Solution

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)

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

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

$space space space space space space space space space equals space space fraction numerator left parenthesis square root of 2 right parenthesis end root to the power of 7 minus 3 left parenthesis square root of 2 right parenthesis to the power of 6 plus 2 left parenthesis square root of 2 right parenthesis end root to the power of 5 minus 5 left parenthesis square root of 2 right parenthesis to the power of 4 plus 4 left parenthesis square root of 2 right parenthesis end root cubed minus 19 left parenthesis square root of 2 right parenthesis end root squared plus 8 left parenthesis square root of 2 right parenthesis end root minus 42 over denominator left parenthesis square root of 2 right parenthesis cubed minus 3 left parenthesis square root of 2 right parenthesis squared plus 2 left parenthesis square root of 2 right parenthesis minus 6 end fraction$
$space space space space space space space space space equals space fraction numerator 8 square root of 2 minus 24 plus 8 square root of 2 minus 20 plus 8 square root of 2 minus 38 plus 8 square root of 2 minus 42 over denominator 2 square root of 2 minus 6 plus 2 square root of 2 minus 6 end fraction space space space space$ $space space space space open square brackets because space straight x space not equal to square root of 2 space rightwards double arrow space straight x squared minus 2 not equal to 0 close square brackets$

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

For Daily Free Study Material Join wiredfaculty

Limits And Derivatives

Some More Questions From Limits and Derivatives Chapter

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Phone Number

Email Address

Loaction