document.write( "Question 391550: How do you find the vertical and horizontal asymptotes of a function? \n" ); document.write( "

Algebra.Com's Answer #278061 by richard1234(7193) $\"\"$ $\"About$

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A vertical asymptote occurs when there exists an x such that f(x) = p/0, where p is nonzero. For example, the function $\"f%28x%29+=+%28x%2B5%29%2F%28x-3%29\"$ has a vertical asymptote at x = 3, since you obtain f(3) = 8/0, but the function $\"f%28x%29+=+%28%28x%2B3%29%28x%2B2%29%29%2F%28x%2B2%29\"$ does not have a vertical asymptote at -2, since the function is equivalent to $\"f%28x%29+=+x%2B3\"$ except at -2 (where a \"gap\" in the function exists). Also, substituting x = -2, we get 0/0 which is indeterminate.\r
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\n" ); document.write( "\n" ); document.write( "On the other hand, a horizontal asymptote occurs when a function converges to a limit as x goes to infinity or negative infinity. In limit notation, a function has a horizontal asymptote if $\"lim%28x-%3Einfinity%2C+f%28x%29%29\"$ exists (or if the limit as x approaches negative infinity exists). There are many types of functions with horizontal asymptotes, and many different ways to find them. \n" ); document.write( "

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