The standard formula of a hyperbola is:
\[\frac{x^2}{a^2} - \frac{y^2}{(c^2 - a^2)} = 1\]
We can rearrange this:
\[\left( \frac{x^2}{a^2} - 1 \right) (c^2 - a^2) = y^2\]
Let define \(b\) such that \(b^2 = c^2 - a^2\):
\[y^2 = b^2(x^2/a^2 - 1)\] \[y = ± b \sqrt{x^2/a^2 - 1}\]
We can multiply the RHS with \(\frac{x}{a} * \frac{a}{x}\):
\[y= ± \left( \frac{xb}{a} \right) \left( \frac{a}{x} \sqrt{\frac{x^2}{a^2}-1}\right)\]
we can simplify this:
\[y = ±\left( \frac{xb}{a} \right) \sqrt{1 - \frac{a^2}{x^2}} \]
If we take the limit \(x \to \infty \), the first term \([xb/a]\) will increase linearly as \(x\) increases, and the second term will approach \(\sqrt{1-0}\). So as \(x\) approaches \(\infty\), the second term will approach \(1\), which means that the hyperbola will continually approach the lines \(±\frac{xb}{a}\), but never touch it. It suffices to say that the asymptote is:
\[y = ±\frac{xb}{a} \]