Hello everyone Originally Posted by
shane99 When will a function cross the slant or horizontal asymptote? Is there a way to tell from a given function that it will cross the asymptote.
Originally Posted by
Prove It By definition, an asymptote does not get crossed...
Sorry, Prove It, but this is not true. Of course an asymptote can be crossed. The definition of an asymptote is (loosely) at tangent to the curve 'at infinity'; but that doesn't preclude the curve from crossing the asymptote at other points.
Consider, for example, the graph of
$\displaystyle y =(x-1)e^{-x}$
As $\displaystyle x \to \infty, y\to 0$. So the $\displaystyle x$-axis is an asymptote.
But when $\displaystyle x = 1, y = 0$. So the curve crosses this asymptote at $\displaystyle (1,0)$.
As far as your original question is concerned, you would just need to look for solutions to an equation in the usual way. If, for example, the graph of $\displaystyle y = f(x)$ has $\displaystyle y = mx+c$ as an asymptote, then the graph crosses the asymptote at any finite values of $\displaystyle x$ that satisfy the equation $\displaystyle f(x) = mx+c$. Of course, there may not be any such values - but that's another issue!
Grandad