The conditions in which a graph can cross through an asymptote depends on what kind of asymptote it is. It can be horizontal, vertical, or undefined. The graph can sometimes cross through a horizontal asymptote, but only through the middle of the graph. Graphs can sometimes cross through a slant asymptote also only through the middle of the graph. Graphs can never cross through a vertical asymptote; it would be undefined.
Example: 
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