document.write( "Question 1197629: What are the vertical and a horizontal asymptote at y, if any, for the function f(x)=x^2+x-30 / x^2-2x-15 \n" ); document.write( "

Algebra.Com's Answer #830982 by greenestamps(13200) $\"\"$ $\"About$

You can put this solution on YOUR website!

\n" ); document.write( "To determine the vertical asymptotes, write the function with numerator and denominator in factored form:

\n" ); document.write( "

\n" ); document.write( "The denominator of the fraction can't be 0, so the function value is not defined at x=5 or at x=-3; those two values of x are the potential vertical asymptotes.

\n" ); document.write( "Since the factor (x-5) is in both numerator and denominator, the function is equivalent to

\n" ); document.write( " $\"%28x%2B6%29%2F%28x%2B3%29\"$

\n" ); document.write( "everywhere except at x=5. So at x=5 there is a hole in the graph instead of a vertical asymptote.

\n" ); document.write( "So there is a vertical asymptote and x=-3, but none at x=5.

\n" ); document.write( "The horizontal asymptote of the function is the function value when x becomes very large positive or very large negative. For very large positive or negative values of x, the linear and constant terms become insignificant, so the function value approaches $\"x%5E2%2Fx%5E2=1\"$

\n" ); document.write( "So the horizontal asymptote is y=1.

\n" ); document.write( "ANSWERS: vertical asymptote x=-3; horizontal asymptote y=1.

\n" ); document.write( "A graph, showing the given function and its horizontal asymptote...

\n" ); document.write( " $\"graph%28800%2C400%2C-20%2C20%2C-10%2C10%2C%28x%5E2%2Bx-30%29%2F%28x%5E2-2x-15%29%2C1%29\"$

\n" ); document.write( "Note the hole at x=5 won't show up on this graph, because the function is undefined only at that one point. Graphing the given function on a very small interval around x=5 on a good graphing calculator will show the hole in the graph.

\n" ); document.write( " \n" ); document.write( "

\n" );