These long term behaviors are very hard to understand.​

the first onethe degree of the polynomial in the numerator is 2.the degree of the polynomial in the denominator is 2.when the top and bottom have the same degree, like in this case, the horizontal asymptotes that that can afford us is simply the value of their coefficients.[tex]\bf \cfrac{x^2-16}{x^2+2x+1}\implies \cfrac{1x^2-16}{1x^2+2x+1}\implies \stackrel{\textit{horizontal asymptote}}{y=\cfrac{1}{1}\implies y=1}[/tex]for the second onewell, the degree of the numerator is 3.the degree of the denominator is 2.when the numerator has a higher degree than the denominator, there are no horizontal asymptotes, however, when the degree of the numerator is exactly 1 degree higher than that of the denominator, the rational has an oblique or slant asymptote, and its equation comes from the quotient of the whole expression, check the picture below, the top part.for the third onethis one is about the same as the one before it, the numerator has exactly one degree higher than the denominator, so we're looking at an oblique asymptote, check the picture below, the bottom part.