You're absolutely right about $f(x)$: no asymptotes there.

For $g(x)$, given your tags, I'm not sure with you've been exposed to the concept of a limit. If you have, simply taking $\lim_{x\to\infty} g(x)$ and you will find $g(x)$ has a horizontal asymptote $y=1$. If you, are unfamiliar with this concept, just remember that $g(x)$ is an exponential decay, so the $24(0.9)^x$ piece will decay to 0 as x gets very large. Therefore, ignoring the $24(0.9)^x$ piece, you find $g(x)$ has a horizontal asymptote $y=1$.

For $h(x)$, you can see it doesn't have a vertical asymptote because the denominator never is 0. Either using limits ($\lim_{x\to-\infty} h(x)$ and $\lim_{x\to-\infty} h(x)$) or graphing over a sufficiently large window, you should be able to convince yourself that as $x$ grows very large (approaching $+\infty$), $h(x)$ approaches 260 and as $x$ approaches $-\infty$, $h(x)$ approaches 0. Therefore, $h(x)$ has two horizontal asymptotes $y=0$ and $y=260$.