Question

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

Solution

Evaluating the function at $WiredFaculty$ it si of $space 0 over 0$ form.
Therefore, $space limit as space space space space space space space space space space space space straight x rightwards arrow square root of 2 of fraction numerator square root of 3 plus 2 straight x end root minus open parentheses square root of 2 plus 1 close parentheses over denominator straight x squared minus 2 end fraction space equals limit as straight x rightwards arrow square root of 2 of fraction numerator square root of 3 plus 2 straight x end root space minus open parentheses square root of 2 plus 1 close parentheses over denominator straight x squared minus 2 end fraction straight x fraction numerator square root of 3 plus 2 straight x end root plus open parentheses square root of 2 plus 1 close parentheses over denominator square root of 3 plus 2 straight x end root plus left parenthesis square root of 2 plus 1 right parenthesis end fraction$

   $WiredFaculty$

$space space space space space space space space space space space equals space space space fraction numerator 2 over denominator left parenthesis square root of 2 plus square root of 2 right parenthesis end root space left parenthesis square root of 3 plus 2 square root of 2 end root plus square root of 2 plus 1 right parenthesis end fraction$
   $WiredFaculty$

   $space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open parentheses because space square root of 3 plus 2 square root of 2 end root equals square root of left parenthesis square root of 2 plus 1 right parenthesis squared end root equals square root of 2 plus 1 close parentheses$

   $space space space space space space space space space space space space space space space space space space equals space fraction numerator 1 over denominator 2 square root of 2 left parenthesis square root of 2 plus 1 right parenthesis end fraction$



   $WiredFaculty$ $WiredFaculty$