Look at the places where each variable approaches either zero or infinity. Look at what happens to the other variables. If you apply this idea to y = 1/x you will see that the y axis is also an asymptote. The graphs of the two asymptotes of y = 1/x are x =0 and y = 0. For another example, consider y = 1/(x - 1). When x =0, y = 0, but in neither case is there an asymptote. When x goes to infinity (more correctly: when x increases without bound) the horizontal line y = 1 is an asymptote, and when x approaches 1, y goes to infinity so the line y = 1 is an asymptote.
Check the type of the equation, because this will sometimes save you the trouble of checking what happens when each variable goes to zero or infinity. Polynomials never have asymptotes. Rational functions always have asymptotes where the denominator goes to zero. Trigonometric functions do not have asymptotes as long as they have only sines and cosines (and no rational expressions) and have an infinite number of vertical asymptotes if functions other sine and cosine are involved. The only conic section that has an asymptote is the hyperbola and the asymptotes are a great help in sketching hyperbolas.
Find the asymptotes of a hyperbola by constructing a rectangle centered at the origin whose dimensions are 2a by 2b, where the a and b come from the equation for the hyperbola: x^2/a^2 - y^2/b^2 = 1. Extend the diagonal bisectors of the rectangle to get the asymptotes. The two halves of the hyperbola are easily sketched as mirror image curves that are tangent to the rectangle and approach the asymptotes. The two halves of the hyperbola intersect the X axis at +a and -a, and the equations for the asymptotes are y = ax/b and y = -ax/b.