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As x → ± ∞, 1/(x - 4) → 0, so g(x) → 2. That means that y = 2 is the horizontal asymptote. Since you cannot divide by 0, there is a vertical asymptote at x = 4.
To fill in the table, just plug in the numbers.
g(-5) = 1/(-5 - 4) + 2 = 17/9 for example.