2.2 The limit of a function  (Page 5/12)

Definition

We define three types of infinite limits .

Infinite limits from the left: Let $f\left(x\right)$ be a function defined at all values in an open interval of the form $\left(b,a\right).$

1. If the values of $f\left(x\right)$ increase without bound as the values of x (where $x<\mathit{\text{a}}\text{)}$ approach the number a , then we say that the limit as x approaches a from the left is positive infinity and we write
   
   $\underset{x\to a^{-}}{\text{lim}}f\left(x\right)=\text{+}\infty .$
2. If the values of $f\left(x\right)$ decrease without bound as the values of x (where $x<\mathit{\text{a}}\text{)}$ approach the number a , then we say that the limit as x approaches a from the left is negative infinity and we write
   
   $\underset{x\to a^{-}}{\text{lim}}f\left(x\right)=\text{−}\infty .$

Infinite limits from the right : Let $f\left(x\right)$ be a function defined at all values in an open interval of the form $\left(a,c\right).$

1. If the values of $f\left(x\right)$ increase without bound as the values of x (where $x>\mathit{\text{a}}\text{)}$ approach the number a , then we say that the limit as x approaches a from the left is positive infinity and we write
   
   $\underset{x\to a^{+}}{\text{lim}}f\left(x\right)=\text{+}\infty .$
2. If the values of $f\left(x\right)$ decrease without bound as the values of x (where $x>\mathit{\text{a}}\text{)}$ approach the number a , then we say that the limit as x approaches a from the left is negative infinity and we write
   
   $\underset{x\to a^{+}}{\text{lim}}f\left(x\right)=\text{−}\infty .$

Two-sided infinite limit: Let $f\left(x\right)$ be defined for all $x\ne a$ in an open interval containing a .

1. If the values of $f\left(x\right)$ increase without bound as the values of x (where $x\ne \mathit{\text{a}}\text{)}$ approach the number a , then we say that the limit as x approaches a is positive infinity and we write
   
   $\underset{x\to a}{\text{lim}}f\left(x\right)=\text{+}\infty .$
2. If the values of $f\left(x\right)$ decrease without bound as the values of x (where $x\ne \mathit{\text{a}}\text{)}$ approach the number a , then we say that the limit as x approaches a is negative infinity and we write
   
   $\underset{x\to a}{\text{lim}}f\left(x\right)=\text{−}\infty .$