# Why is it impossible to have lim_(x->0) f(x) and lim_(f(x)->0)f(x) simultaneously exist for any of these graphs?

## A) $f \left(x\right) = \frac{1}{x} ^ 2$ B) $f \left(x\right) = - \frac{1}{x} ^ 2$ C) $f \left(x\right) = \frac{1}{x}$ D) $f \left(x\right) = - \frac{1}{x}$

Feb 12, 2017

All four of these graphs have the $x$-axis as a horizontal asymptote as $x \to \pm \infty$ and the $y$-axis as a vertical asymptote as $x \to 0$ from the right or left.

Feb 12, 2017

Well, by definition, a vertical asymptote is when at $x \to a$, $y \to \pm \infty$ from either side and $x$ never touches $a$. Similarly, a horizontal asymptote is when at $x \to \pm \infty$, $y \to b$ from either side without ever reaching $b$.

For the function

$y = \frac{c}{x}$,

where $c$ is a constant, if $x \to 0$, $y \to \pm \infty$ from either side of $x = 0$, so you have a vertical asymptote. You can also find that as $x \to \pm \infty$, $y \to 0$ but doesn't get there.

But if you have $x \to 0$ and consequently $y \to \pm \infty$, you can't also have $x \to \pm \infty$ so that $y \to 0$. It's not possible to approach both asymptotes at once because $x$ cannot approach $0$ and $\pm \infty$ at the same time, and $y$ cannot approach $\pm \infty$ and $0$ at the same time.

(Imagine trying to run to two different places at once; can't do it.)

Both kinds of asymptotes are on the graph, to be sure, but you can only approach one of those kinds of asymptotes at a time.