Epsilon-Delta Limits

\textrm{Definition of a limit: Let } f \textrm{ be a function defined at all points of an open} \\ \textrm{interval } I \textrm{ containing } c \textrm{ except perhaps at } c \textrm{ itself. We say } \lim_{x \rightarrow c} f(x) = L \textrm{ for} \\ \textrm{some real number } L \textrm{ if for every } \epsilon > 0 \textrm{ there exists a } \delta > 0 \textrm{ such that} \\ |f(x) - L| < \epsilon \textrm{ whenever } x \textrm{ is in } I \textrm{ and satisfies } 0 < |x-c| < \delta.