x²+8x+7=(x+1)(x+7), when x=0, this expression=7.
x²+11x+30=(x+6)(x+5), when x=0, this expression=30.
f(x)=(x+1)(x+7)/((x+6)(x+5)) (red).
The y-intercept=7/30 (0,0.2333).
The x-intercepts are x=-1 and -7 (zeroes of the numerator).
The vertical asymptotes (green) are x=-5 and -6 because these values make the denominator zero.
The horizontal asymptote (blue) is x²/x²=1, so it’s y=1, because, when x is very large (positive or negative), all terms apart from x² can be ignored as insignificant.