document.write( "Question 1039497: lim (x, y) -> (0, 0) $\"+%28xy+%2B+y%5E3%29+%2F+%28x%5E2+%2B+y%5E2%29+\"$ \r
\n" ); document.write( "\n" ); document.write( "I am having a hard time knowing how to solve this properly.
\n" ); document.write( "Simply plugging in the values doesnt work.
\n" ); document.write( "Neither does factoring.\r
\n" ); document.write( "\n" ); document.write( "Which lead me to try y=0 [then solve], and x=0 [then solve].\r
\n" ); document.write( "\n" ); document.write( "if $\"+y+=+0+\"$ , then I end up with $\"+0+%2F+x%5E2+\"$
\n" ); document.write( "if $\"+x+=+0+\"$ , then I end up with $\"+y+\"$ \r
\n" ); document.write( "\n" ); document.write( "Because the two answers dont match up, I am led to believe that the limit does not exist (DNE).\r
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\n" ); document.write( "Then, as I was looking through my notes I noticed another method that my teacher suggested we try. It involves transforming the polar coordinates (replacing $\"+x+\"$ with $\"+r+cos+theta+\"$ and replacing $\"+y+\"$ with $\"+r+sin+theta+\"$ \r
\n" ); document.write( "\n" ); document.write( "Upon doing this I ended up with:
\n" ); document.write( "lim r-> 0
\n" ); document.write( "r (cos sin theta) + (sin^3 theta) = 0\r
\n" ); document.write( "\n" ); document.write( "So does the limit exist? Or is it zero? \n" ); document.write( "

Algebra.Com's Answer #654256 by Edwin McCravy(20059) $\"\"$ $\"About$

You can put this solution on YOUR website!

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document.write( "The limit may very well not exist.  Always test for this\r\n" );
document.write( "possibility first.\r\n" );
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document.write( "Let's test for this:\r\n" );
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document.write( "If the value of the limit is not the same for all possible \r\n" );
document.write( "approaches, or paths, to (0,0), then the limit does not exist.\r\n" );
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document.write( "Let's choose the path y=kx for some constant k.\r\n" );
document.write( "So we substitute kx for y:\r\n" );
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document.write( "When x approaches 0, this approaches\r\n" );
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document.write( "  \r\n" );
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document.write( "which has different values for different choices of k.\r\n" );
document.write( "For instance, it's 0 if k=0 and 1/2 if k=1, and 2/5 if k=2, etc.\r\n" );
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document.write( "That means the limit cannot exist, for it must approach\r\n" );
document.write( "the same value regardless of the path taken to the origin.\r\n" );
document.write( "But as we see this is not the case.\r\n" );
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document.write( "Answer:  the limit does not exist.\r\n" );
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document.write( "Edwin

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