## Precalculus (6th Edition) Blitzer

From $c^2=a^2+b^2$, dividing both sides by $a^2$, we have $\frac{c^2}{a^2}=1+\frac{b^2}{a^2}$. When $\frac{c}{a}\to\infty$, we have $\frac{b}{a}\to\infty$. Thus the slope of the asymptote will be $\frac{b}{a}\to\infty$, indicating nearly vertical asymptotes and hyperbola branches.