**Introduction**

\lim_{x \rightarrow c}f(x) is read as the limit of f(x) as x approaches to c is L .

**When do we use \lim_{x \rightarrow c}f(x) ?**

Consider the function f(x)=\frac{x-1}{x-1} . We know that the function is undefined when x=1 . In cases like this, we are interested on the behavior of the function as x approaches to 1 . To examine the behavior of the function as x approaches to 1 , we consider \lim_{x \rightarrow 1}\frac{x-1}{x-1} .

**What does \lim_{x \rightarrow 1}\frac{x-1}{x-1} mean?**

\lim_{x \rightarrow 1}\frac{x-1}{x-1} mean that we are studying the behavior of the function as x approaches to 1 from the left of 1 and the behavior of the function as x approaches to 1 from the right of 1 . In doing this so, we can use table of values to study such behavior.

Let’s start studying the behavior of the function as x approaches to 1 from the left of 1 , then we’re dealing with \lim_{x \rightarrow 1^{-}}\frac{x-1}{x-1} . The arrow pointing at 1 indicates that x is approaching 1 and the superscript “ - ” indicates that x is approaching 1 from the left. From this definition, we can construct table of values:

x | 0 | 0.9 | 0.99 | 0.999 | 0.9999 |

f(x) | 1 | 1 | 1 | 1 | 1 |

*As x approaches to 1 from the left, the function remains the same all throughout. Hence, we say that \lim_{x \rightarrow 1^{-}}\frac{x-1}{x-1}=1 . This limit is also called the left hand limit.*

Now, let’s study the behavior of the function as x approaches to 1 from the right of 1 , then we’re dealing with \lim_{x \rightarrow 1^{+}}\frac{x-1}{x-1} . The arrow pointing at 1 indicates that x is approaching 1 and the superscript “ + ” indicates that x is approaching 1 from the right. From this definition, the table of values would be:

x | 2 | 1.1 | 1.01 | 1.001 | 1.0001 |

f(x) | 1 | 1 | 1 | 1 | 1 |

*As x approaches to 1 from the right, the function remains the same all throughout. Hence, we say that \lim_{x \rightarrow 1^{+}}\frac{x-1}{x-1}=1 . This limit is called the right hand limit.*

**Definition of the Limit**

For any function f , \lim_{x \rightarrow c}f(x)=L exists if and only if the left hand limit \lim_{x \rightarrow c^{-}}f(x) is equal to the right hand limit \lim_{x \rightarrow c^{+}}f(x) . In other words, if \lim_{x \rightarrow c^{-}}f(x)=L and \lim_{x \rightarrow c^{+}}f(x)=L , then \lim_{x \rightarrow c}f(x)=L .

The \lim_{x \rightarrow 1}\frac{x-1}{x-1}=1 because \lim_{x \rightarrow 1^{-}}\frac{x-1}{x-1}=1 and \lim_{x \rightarrow 1^{+}}\frac{x-1}{x-1}=1 .

**So now, what’s the difference between the \lim_{x \rightarrow c}f(x) and f(c) ?**

\lim_{x \rightarrow c}f(x) and f(c) denote two different concepts such that the former deals with the limit of the function as x approaches to a particular number c and the latter is just basically the functional value at which x=c . In the function f(x)=\frac{x-1}{x-1} , the \lim_{x \rightarrow 1}\frac{x-1}{x-1}=1 while the functional value at which x=1 is f(1)=\frac{1-1}{1-1}=\frac{0}{0} which is undefined.

**Graph of the Function**In this function, it’s easy to conclude that whatever the value of x, our function remains the same. However, the function is undefined when x is equal to 1. Hence, it leaves a hole in its graph.