Limits and Their Properties

The idea of limits is fundamental to the study of calculus. It suggests a boundary that may be reached or not, or possibly exceeded. A mathematical limit has characteristics similar to those of a physical limit. It is the analysis of how function values or outputs change, when inputs change.

In symbols, we write this process as
 $\lim_{x\rightarrow c}\ f\left(x\right)=L$  
This is read, ‘‘The limit of f(x) as x approaches c is L.”

Evaluate the  $\lim_{x\rightarrow2}\left(1+3x\right)$  using the table of values

Imagine a number line, Substitute any number close to 2 from the left

Let's say x=1.9, then  $\lim_{x\rightarrow2}\left(1+3x\right)=6.7$  
We get the another number closer than the first still from the left of 2

The limits of the function exists if
(i) the  $\lim_{x\rightarrow c^{\ -\ }}f\left(x\right)$  exists