Asymptote is a straight line which passes very close to curve till infinite but never intersects the curve. Asymptote is mainly of three types, (1) Horizontal Asymptotes (2) Vertical asymptotes (3) oblique asymptotes. Suppose p = f(q) is a function so any line p = a, is called its horizontal asymptotes only when following condition is true:- Limit q -> +- ∞ f(q) = a.
Now rules for asymptotes are:-
Rule 1: In any function if numerator is greater than denominator then in this case there is not any horizontal asymptotes. But if numerator is greater than denominator by degree one then in this case asymptotes will be oblique asymptotes or slant asymptotes.
Suppose there is a function: f (x) = 4x^{3} + 4x+1 / x^{2} + 1,
Limit x-> +∞ f(x) = ∞;
Limit x-> -∞ f(x) = -∞;
In this case we can find oblique asymptotes of long division method.
Rule 2: In any function if degree of numerator and denominator are equal then in this case we can find horizontal asymptotes by fraction of leading coefficient. Horizontal asymptotes of function f(x) = (8 x^{3} + 4 x + 1) / (2x^{3} + 6) is, y = 8 /2=> y = 4.
Horizontal asymptotes =4,
Rule 3: If degree of numerator is less than degree of denominator than in this case horizontal asymptotes will be a line, y = 0.
F(x) = 2x^{2} + 3x / 2x^{4} + 6; so horizontal asymptote for this function, y = 0.
Rule 4: Vertical asymptote rules: For vertical asymptotes we will put denominator equals to 0, and find value of 'x'.