I was taught that an asymptote cannot be crossed. My teacher then went and made my life a lot harder by countering what I've learned. Why can some asymptotes be crossed?
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$\begingroup$The definition on Wikipedia is:
A line $L$ is said to be an asymptote if the distance between the curve and the line becomes zero as $t$ tends to $\infty$.
Using this definition even if the curve crosses and cuts the asymptote but if its distance from the curve decreases then it is still an asymptote.
$\endgroup$ $\begingroup$The definition of horizontal asymptote is that $f:\mathbb{R} \to \mathbb{R}$ has a horizontal asymptote $L$ if $\lim_{x \to \pm\infty} f(x) = L$. (Or similar definition for asymptote at just $+\infty$ or just $-\infty$). There is nothing in this definition that requires that the asymptote cannot be crossed.
What your teacher may have been referring to is a vertical asymptote, for which the definition is that $f$ has a vertical asymptote at $x=a$ if $\lim_{x \to a}|f(x)| = \infty$. In this case $f$ cannot "cross" the vertical asymptote because if it crossed it, then $f(a)$ would be defined and $|f(x)|$ would not tend to $\infty$ as $x \to a$ (assuming $f$ is continuous on its domain). Although it is possible that this is what your teacher meant, it is still a very shaky thing to say, and I would be wary not to just believe what he or she tells you without proof.
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