TITLE: Where does the equation of asymptotes of a hyperbola come from?
QUESTION [10 upvotes]: It's known that the asymptotes of a hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ is given by $y=\pm\frac{b}{a}x$ if $a>b$.
I  tried to find a proof of the fact that why the equations of these asymptotes are like that,however the only reference (Thomas calculus  book) that I found explained 
that the two asymptotes are derived by letting $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=0$.

It would be highly appreciated if someone prove why the equation of the asymptotes have such form.

REPLY [6 votes]: The other answers have tried to give more rigorous arguments, which I would like to complement with a heuristic way which doesn't need further manipulation of the equation:
From looking at the hyperbola, it is obvious that the asymptotes are lines that the curve approaches when $x$ and $y$ become very large, in particular larger than $a$ or $b$ (BTW, the condition $a<b$ seems unnecessary to me). Then, in the defining equation$$\frac{x²}{a²}-\frac{y²}{b²}=1\,,$$ you have two large numbers on the left-hand side whose difference is $1$. In other words, their difference is much smaller than the numbers themselves, and it becomes a good approximation to just neglect the $1$ on the right-hand side. Furthermore, the approximation becomes better which increasing $x$ and $y$. Thus, $$\frac{x²}{a²}-\frac{y²}{b²}=0\,,$$ is at least a good candidate for the equation of the asymptotes.