A continuous real-valued function $f: A \to B$, such that $A, B \subseteq \mathbb{R}$ where $f(0)=1$ and $\displaystyle \lim_{x \to \infty} f(x)=-1$ must have a zero within $(0,\infty)$
However, if an asymptote existed at some $x = \alpha$, where $\alpha \notin A$ such that $f(x) \to \infty$ as $x \to \alpha^{+}$ and $f(x) \to -\infty$ as $x \to \alpha^{-}$, and in the interval $(\alpha, \infty)$, the function approaches the limit $f(x) = -1$ as $x \to \infty$, isn't there a possibility that a zero does not exist within $(0,\infty)$, thus disproving the statement?
EDIT: Also, what if $\alpha \notin A$ such that $f(x) \to \infty$ as $x \to \alpha^{-}$ and $f(x) \to -\infty$ as $x \to \alpha^{+}$? Sorry for the edit again.