Definition of an Infinite Limit

Let $f$ be a function defined on an open interval containing $c$ (except possibly at $c$ .) The statement

$$\lim_{x\to c} f(x)=\infty$$

means that for each $M>0$ there exists a $\delta >0$ such that

$$f(x)>M \quad 0<\left\lvert x-c \right\rvert <\delta.$$

Similarly, the statement

$$\lim_{x\to c} f(x)= -\infty$$

means that for each $N<0$ there exists a $\delta >0$ such that

$$f(x)<N \quad 0<\left\lvert x-c \right\rvert <\delta.$$

To define the infinite limit from the left, replace $0<\left\lvert x-c \right\rvert <\delta$ by $c-\delta<x<c.$ To define the infinite limit from the right, replace $0<\left\lvert x-c \right\rvert <\delta$ by $c<x<c+\delta.$