Question with asymptote with function

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I have a function $$\frac{x}{x^2+1}$$ and have determined the horizontal asymptote to be $y = 0$ because the denominator has the highest degree. However when I graph this function $y$ is $0$ at the origin. Both the left and right hand side of the graph do have the asymptote. Why does the asymptote not respected at the origin?