document.write( "Question 1044892: Kindly help me solving it i tried my best but I couldn't
\n" ); document.write( "lim \"%28x%5E%285%2F2%29-2%5E%285%2F4%29%29%2F%28sqrt+%28x%29-2%5E%281%2F4%29%29\" is equal to
\n" ); document.write( "x➡\"sqrt+%282%29\"
\n" ); document.write( "a)1/10
\n" ); document.write( "b)10
\n" ); document.write( "c)20
\n" ); document.write( "d)none of these \n" ); document.write( "
Algebra.Com's Answer #660322 by KMST(5328)\"\" \"About 
You can put this solution on YOUR website!
Your expression/function of \"x\" is
\n" ); document.write( "\"f%28x%29=%28x%5E%225%2F2%22-2%5E%225%2F4%22%29%2F%28sqrt+%28x%29-2%5E%221%2F4%22%29\"\"%22=%22\"\"%28x%5E%225%2F2%22-2%5E%225%2F4%22%29%2F%28x%5E%221%2F2%22-2%5E%221%2F4%22%29\"\"%22=%22\" .
\n" ); document.write( "Looks complicated, but it can be simplified.
\n" ); document.write( "We need a change of variable, so we do not have to struggle with those fractional exponents.
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\n" ); document.write( "If I rewrite your function with \"x%5E%221%2F2%22=a\" ,
\n" ); document.write( "and define a constant \"b=2%5E%221%2F4%22\" ,
\n" ); document.write( "\"f%28x%29\" looks as simple as \"g%28a%29=%28a%5E5-b%5E5%29%2F%28a-b%29\" , a rational function of \"a\" ,
\n" ); document.write( "with the constant \"b%7D%7D+such+that+%7B%7Bb=2%5E%221%2F4%22\"-->\"system%28b%5E2=2%5E%221%2F2%22=sqrt%282%29%2C%22and%22%2Cb%5E4=2%29\" .
\n" ); document.write( "Now, if \"a%3C%3Eb\" <---> \"a-b%3C%3E0\" ,
\n" ); document.write( "we know a way to transform that rational function into a polynomial:
\n" ); document.write( "\"g%28a%29=%28a%5E5-b%5E5%29%2F%28a-b%29=a%5E4%2Ba%5E3b%2Ba%5E2b%5E2%2Bab%5E3%2Bb%5E4\" for \"a%3C%3Eb\" .
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\n" ); document.write( "Now \"f%28x%29=g%28a%29\" , and \"%22=%22\"
\n" ); document.write( "\"lim%28x-%3Esqrt%282%29%2Cf%28x%29%29\"\"%22=%22\"\"lim%28a-%3Eb%2Cg%28a%29%29\"\"%22=%22\"\"g%28b%29\"\"%22=%22\"\"%28b%5E4%2Bb%5E3b%2Bb%5E2b%5E2%2Bb%29b%5E3%2Bb%5E4\"\"%22=%22\"\"b%5E4%2Bb%5E4%2Bb%5E4b4%2Bb%5E4%2Bb%5E4\"\"%22=%22\"\"5b%5E4=5%2A2=highlight%2810%29\"\r
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\n" ); document.write( "\n" ); document.write( "NOTES:
\n" ); document.write( "1) Your instructor may favor \"u\" rather than \"a\" as the variable for a change of variable.
\n" ); document.write( "I chose \"a\" because it should remind you of what you learned in algebra and/or \"pre-calculus\".
\n" ); document.write( "2) You may remember from algebra/pre-calculus that
\n" ); document.write( "\"a%5E2-b%5E2=%28a%2Bb%29%28a-b%29\" , and
\n" ); document.write( "\"a%5E3-b%5E3=%28a%5E2%2Bab%2Bb%5E2%29%28a-b%29\" .
\n" ); document.write( "You should not be surprised to find out that
\n" ); document.write( "\"a%5E5-b%5E5=%28a%5E4%2Ba%5E3b%2Ba%5E2b%5E2%2Bab%5E3%2Bb%5E4%29%28a-b%29\" <---> \"%28a%5E5-b%5E5%29%2F%28a-b%29=a%5E4%2Ba%5E3b%2Ba%5E2b%5E2%2Bab%5E3%2Bb%5E4\" .
\n" ); document.write( "Alternately, you may recognize \"a%5E4%2Ba%5E3b%2Ba%5E2b%5E2%2Bab%5E3%2Bb%5E4\"
\n" ); document.write( "as the sum of the first five terms of a geometric sequence
\n" ); document.write( "with first term \"a%5E4\" and common ratio \"r=b%2Fa\" ,
\n" ); document.write( "and that sum would be \n" ); document.write( "
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