Find vertical asymptotes by setting the denominator equal to zero and checking where the function is undefined
Factor and simplify first to cancel any removable discontinuities
Find horizontal asymptotes by comparing the degrees of the numerator and denominator
If the numerator degree is less than the denominator degree, the horizontal asymptote is y = 0
If the numerator degree equals the denominator degree, the horizontal asymptote is the ratio of leading coefficients
If the numerator degree is greater than the denominator degree, there is no horizontal asymptote
Find slant asymptotes by dividing the numerator by the denominator when the numerator degree is exactly one more than the denominator degree
Use polynomial long division or synthetic division to get the slant asymptote
For non-rational functions, analyze limits as x approaches values that make the function undefined
Check end behavior by evaluating limits as x approaches positive infinity and negative infinity
Verify asymptotes by confirming the function approaches the line but does not equal it over the relevant interval