document.write( "Question 1117348: Without using L'Hospitals rule find :
\n" ); document.write( " Limt((e^(17x)-17e^x +16)/(e^(16x)-16e^x+15)). as x=0 \n" ); document.write( "

Algebra.Com's Answer #732410 by ikleyn(52818) $\"\"$ $\"About$

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document.write( "Introduce  z = .  Then the given fraction takes the form\r\n" );
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document.write( "    fraction = .    (1)\r\n" );
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document.write( "The polynomial    has the root z=1  and therefore is divided by (z-1) without a remainder.\r\n" );
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document.write( "The factoring formula is  \r\n" );
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document.write( " = .     (2)\r\n" );
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document.write( "Similarly, the polynomial    has the root z=1  and therefore is divided by (z-1) without a remainder.\r\n" );
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document.write( "The factoring formula is  \r\n" );
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document.write( " = .     (3)\r\n" );
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document.write( "If you substitute (2) and (3)  into (1), you will get after canceling (z-1)\r\n" );
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document.write( "    fraction =     (4)\r\n" );
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document.write( "It is still not a safe situation, since both polynomials in numerator and denominator of (4) have z= 1 as a root.\r\n" );
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document.write( "So, we need divide each of (2) and (3) by (z-1) one more time. If you do it, you will get\r\n" );
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document.write( " = ,   (5)\r\n" );
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document.write( " = .   (6)\r\n" );
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document.write( "Hence, when you substitute (5) and (6) into (4) and cancel the common factor (z-1) again, you will get\r\n" );
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document.write( "    fraction =    (7)\r\n" );
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document.write( "Now you can safely find the limit of (7) at z ---> 1  simply substituting z = 1 into its numerator and denominator. You will get\r\n" );
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document.write( "    fraction limit at z --> 1 is equal to  =    (8)\r\n" );
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document.write( "Easy summation of arithmetic progressions gives  Numerator =  = 136,  Denominator =  = 120.\r\n" );
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document.write( "Hence the answer is:  The given fraction limit at x ---> 0 is    = .\r\n" );
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