Print value of vertical & horizontal asymptote?

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When you write:

solve( 7/(x-3) == 0, x)

you look for solutions of the equation $7/(x-3) = 0$, which has no solution. If you want to discover the poles of your function (not the zeroes) to locate where your function has vertical asymptotes, you should look for the zeroes of the inverse of your function:

sage: f(x) = 7/(x-3)
sage: ans = solve(1/f == 0, x)
sage: ans[0].rhs()
3

If you want to locate the horizontal asymptotes, you shoult look at the limit of your function in $+\infty$ and $-\infty$:

sage: limit(f,x=+infinity)
x |--> 0
sage: limit(f,x=-infinity)
x |--> 0

Here, the vertical asymptote has equation $x=3$, and the horizontal asymptote has equation $y=0$.