Math Help - Confusion in sketching this graph

1. Confusion in sketching this graph

$\displaystyle f(x)=\frac{x-1}{x^2+3}$

There appears to be an x-intercept at $x=1$, but I thought there is an asymptote at the x-axis since the denominator has a greater degree than the numerator?

2. Whether the x-axis is an asymptote describes the behavior of the function as x approaches infinity, and whether there is an x-intercept at x = 1 describes the behavior around x = 1. Both can be true.

3. There is a horizontal asymptote at y= 1. There are no vertical asmptytotes because the denominator is never 0.

4. Isn't horizontal asymptote at y = 0?

5. Yes, of course. That was a typo. (That's my story and I'm sticking to it!)