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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), you should look for the zeroes of 1/f:

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

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), zeroes) to locate where your function has vertical asymptote, you should look for the zeroes of 1/f:

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

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

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 asymptote, you should look for the zeroes of 1/f:

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 asymptote, 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

When you write, 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 asymptote, asymptotes, you should look for the zeroes of 1/f: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 asymptote, 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 Here, the vertical asymptote has equation $x=3$, and the horizontal asymptote has equation $y=0$.