Limit of Functions

1. Convergence and Divergence of a Function
1. When the value of approaches but is not equal to , if the value of the function approaches a certain value , we say that the function converges to . In this case, is called the limit value or limit of the function as approaches . This is expressed symbolically as:
   or when ,
   
   
2. If the function does not converge to any value, we say that the function diverges. When the value of approaches but is not equal to ,
1. If the value of becomes infinitely large, we say that the function diverges to positive infinity. This is expressed symbolically as:
   or when ,
2. If the value of is negative and its absolute value becomes infinitely large, we say that the function diverges to negative infinity. This is expressed symbolically as:
   or when ,


2. Right-Hand and Left-Hand Limits
   If the limit value of the function at is , then both the right-hand limit and the left-hand limit at exist, and both are equal to . Conversely, the following is true:
   
   On the other hand, even if both the right-hand limit and left-hand limit of the function at exist, if their values are not equal, the overall limit does not exist.