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\n" ); document.write( "The vertical asymptotes will occur for x values that make the denominator zero (and are not \"holes\"). To find them, factor the numerator and denominator:
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\n" ); document.write( "If there was a common factor between the numerator and denominator, then there would be a \"hole\". We do not have a common factor so there are no holes. Now we can proceed to determine the vertical asymptotes. We set the denominator equal to zero:
\n" ); document.write( "(5x-1)(3x+1) = 0
\n" ); document.write( "and solve for x. From the Zero Product Property:
\n" ); document.write( "5x-1 = 0 or 3x+1 = 0
\n" ); document.write( "Solving we get:
\n" ); document.write( "x = 1/5 or x = -1/3
\n" ); document.write( "These are the equations for the vertical asymptotes.
\n" ); document.write( "For the horizontal asymptotes, we start with the original function:
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\n" ); document.write( "and divide each term by the highest power of x. This would be
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\n" ); document.write( "Now we look at this and figure out what happens when x becomes larger and larger (both positively and negatively). As x becomes large each of those fractions (which have an x in the denominator and a constant in the numerator) will become smaller and smaller. IOW: closer and closer to zero. So the numerator will become close to zero the the denominator will become close to 15. 0/15 is zero so the horizontal asymptote is y = 0.
\n" ); document.write( "FWIW: Here's what the graph looks like:
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