> [!NOTE] Definition (Properly divergent function) > Let $I$ be an [[Open Real Interval|open real interval]]. Let $c\in I.$ Let $f$ be a [[Real Function|real function]] defined on $I,$ except possibly at $c.$ We write $\lim_{ x \to c} f(x) = \infty$ iff for every $M>0,$ there exists $\delta>0$ such that for all $x\in I,$ $0<|x-c|<\delta \implies f(x)>M.$ > > We write $\lim_{ x \to c }=-\infty$ iff for every $M<0,$ there exists $\delta>0$ such that for all $x\in I,$ $0<|x-c|<\delta \implies f(x)<M.$