Functions, Graphs, and Limits
Vertical Asymptotes vs. Holes
Both vertical asymptotes and holes are places where the curve never quite touches. Holes occur at places where the limit of the function exists but the function itself does not. These correspond to the roots (or zeros) of the denominator that cancel out entirely during simplification. Vertical asymptotes occur at places where the limit of the function is ∞ or -∞, which happen at the roots of the denominator that are left over after simplification.