Limit of a Function

 

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.

 

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