There are 3 cases to consider when determining horizontal asymptotes:

1) **Case 1**:

if: degree of numerator < degree of denominator

then: horizontal asymptote: y = 0 (x-axis)

$i.e. f(x) = \frac{ax^{3}+......}{bx^{5}+......}$→ horizontal asymptote: $y = 0$

2) **Case 2**:

if: degree of numerator = degree of denominator

then: horizontal asymptote: y = $\frac{leading\; coefficient \;of\; numerator}{leading\; coefficient\; of\; denominator}$

$i.e. f(x) = \frac{ax^{5}+......}{bx^{5}+......}$→ horizontal asymptote: $y = \frac{a}{b}$

3) **Case 3**:

if: degree of numerator > degree of denominator

then: horizontal asymptote: NONE

$i.e. f(x) = \frac{ax^{5}+......}{bx^{3}+......}$→$NO\; horizontal\; asymptote$