Are There Any 'Horizontal Asymptote' Rule Exceptions?
$\Begingroup$ an Equation I Have Is $$F(X) = \Frac{9X(X-9)}{3X^2-11X-4}. $$ Upon Calculating Using the Rules Taught in Class, There Is an H. A. at $Y = 3$ and...
An equation I have is $$F(x) = \frac{9x(x-9)}{3x^2-11x-4}.$$
Upon calculating using the rules taught in class,
There is an H.A. at $y = 3$ and a V.A. at $x = -\frac13$ and at $4.$
After graphing, V.A. seems to be correct, but if I plug in $23/11$ as the $x$ value, it still returns $3$.
Are there any exceptions to when to use the rule to find out H.A.?
2 Answers
Remember that an horizontal asymptote can cross the curve (there are even examples when this happens infinitely many times). The horizontal asymptote only describes the curve's behaviour for $x\to\pm \infty$.
$$y=\lim _{ x\rightarrow +\infty }{ \frac { 9x\left( x-9 \right) }{ 3{ x }^{ 2 }-11x-4 } = } \lim _{ x\rightarrow +\infty }{ \frac { { x }^{ 2 }\left( 9-\frac { 81 }{ x } \right) }{ { x }^{ 2 }\left( 3-\frac { 11 }{ x } -\frac { 4 }{ { x }^{ 2 } } \right) } = } 3$$ so $$y=3$$ is your horizontal asymptote
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