The equation becomes undefined at x=3 in the denominator as 3-3=0. Basically this means that mathematically you are not allowed to divide by 0.
color(blue)("Vertical asymptotes")
lim_(xto3^(+))(x+3)/(x-3) -> (x+3)/(0^+)=+oo
lim_(Xto3^-)(x+3)/(x-3)-> (x+3)/(0^-)=-oo
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color(blue)("Horizontal asymptotes")
As x becomes bigger and bigger then the addition or subtraction of 3 becomes insignificant. Consequently we end up with basically x/x
lim_(xtooo^+) (x+3)/(x-3) ->(+oo)/(+oo) = +1
lim_(xtooo^-) (x+3)/(x-3)->(-oo)/(-oo) = +1
So the horizontal asymptote is +1
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Check: using polynomial division.
(x+3)-:(x-3) = 1+6/(x-3)
lim_(x->3^(+-) ) = 1+-oo
lim_(x->oo^(+-) ) = 1+-0=1