Oblique asymptote question

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First part is original equation, second is after it was divided using polynomial division to find equation for oblique asymptote.I don't understand the 'x-1' part being the equation of the asymptote here,and the rest can be ignored because its basically 0, but for it to be 0 you set x to be nearing infinity, so then doesnt the 'x -1' at the beginning basically = infinity. And then your equation is y = infinity

$$y= \frac{(x^2)}{x+1} , y= x - 1 + \frac{(1)}{x+1}$$

horizontal asymptotes are straight so y is always one value. If I were to plot the graph for the first equation some points on the oblique asymptote would be around like (1,-1/2) so what does the line even mean/show then.I saw someone say to think of

$y = x - 1 + \frac{(1)}{x+1}$

as a function and

$x+1$

as another then compare the two and so on but I dont get why I would do that.