Let's suppose I have the graph y = $\frac 1x$ and that I do not know how it looks visually/graphically. I know there is an asymptote at $x = 0$, but do not know if the graph is approaching positive or negative infinity at both sides of the asymptote. I would like to know if it is possible to tell algebraically just from a given equation where the graph goes on both sides of the asymptote. I would like to do this without plugging in two values for $x$ (smaller than $x$ /greater than $x$), unless that that is the only solution to this problem.
[Math] better way to tell if a function is approaching positive or negative infinity without looking at the graph
algebra-precalculuscalculusfunctions
Best Answer
So you know the limit on one side $\to \infty$ or $\to -\infty$ and would like to figure which one the limit tends to from the two.
This is easy just note $\frac{\text{positive}}{\text{negative}}=\text{negative}$, etc. All those basic rules come in handy.
It all depends on the function though.
Say we want to find $\lim_{\to 0^+}$
$$\frac{1}{x}$$
I think you'll agree the limit is infinite. We want to see what happens as $x>0$ but $x$ gets close to zero. Well because $x$ is positive in that case.
$$\frac{1}{x}$$
Must be also positive.
So it all just had to do with basic rules and often algebraic manipulation.
More examples,
$$\frac{1}{x^2}$$
Find the limit as $x \to 0$
$x^2>0$ so $\frac{1}{x^2}$ is positive. It diverges, so it goes to $+\infty$.
$$-\frac{1}{x^3}$$
$$\to 0^-$$
We approaching from $x<0 \implies x^3<0 \implies -\frac{1}{x^3}>0$. The limit diverges, so it must go to $+\infty$.