Identify the function type: rational, logarithmic, exponential, trigonometric, or other
For vertical asymptotes, find values of x that make the denominator zero after simplifying
Check whether any factors cancel before declaring vertical asymptotes
For horizontal asymptotes, compare degrees of numerator and denominator in rational functions
If numerator degree is less than denominator degree, horizontal asymptote is y = 0
If numerator degree equals denominator degree, horizontal asymptote is the ratio of leading coefficients
If numerator degree is greater than denominator degree, there is no horizontal asymptote
For slant asymptotes, divide the numerator by the denominator when the numerator degree is exactly one more than the denominator degree
The quotient from polynomial division gives the slant asymptote
For oblique or higher-degree asymptotes, use polynomial division when appropriate
For limits, compute lim as x approaches the value or infinity to confirm asymptotes
For logarithmic functions, set the argument equal to zero to find vertical asymptotes
For exponential functions, check end behavior to find horizontal asymptotes
For trigonometric functions, identify values where the function is undefined for vertical asymptotes
Verify asymptotes by graphing or evaluating limits near the suspected asymptote points