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If a variable $x$ takes values which are more and more close to a finite number $a$ , then we say that $x$ approaches $a$ written as $x\to a$).

• If values of $x$ come closer to $a$ but are always greater than $a$ , then we say that $x$ approaches $a$ form right ($x\to a^+$).
• If values of $x$ come closer to $a$ but are always less than $a$ , then $x$ approaches $a$ from left ($x\to a^-$) .

The concept of a limit is essentially what separates the field of calculus, and analysis in general, from other fields of mathematics such as geometry or algebra.

The concept of a limit may apply to:

## Examples

If $x\to2$ , then $x$ can approach to '2' from two sides:

• From right side: In notation we write $x\to2^+$ means $x$ is coming closer to '2' from right i.e. it is more than '2'.
\begin{align}&x=2.1\\&x=2.01\\&x=2.0001\\&\vdots\end{align}
• From left side: In notation we write $x\to2^-$ mean $x$ is coming closer to 2 from left i.e. it is less than '2'.
\begin{align}&x=1.9\\&x=1.99\\&x=1.9999\\&\vdots\end{align}

## Meaning of a limiting value

Let $f(x)$ be function of $x$ . If the expression $f(x)$ comes close to $L$ as $x$ approaches $a$ then we say that $L$ is the limit of $f(x)$ as $x$ approaches $a$ .

In notation, it is written as $\lim_{x\to a}f(x)=L$ .

## Right Hand Limit

If $f(x)$ approaches $L_1$ as $x$ approaches $a$ from the right, then $L_1$ is called as the right hand limit of $f(x)$ .

Right hand limit can be expressed in two ways:-

• $\lim_{x\to a^+}f(x)=L_1$
• $\lim_{h\to0}f(a+h)\text{ Put }x=a+h\text{ in the above result}$

## Left Hand Limit

If $f(x)$ approaches $a$from the left, then $L_2$ is called the left hand limit of $f(x)$ . The left hand limit can be expressed in two ways:-

• $\lim_{x\to a^-}f(x)=L_2$
• $\lim_{h\to0}f(a-h)\text{ Put }x=a-h\text{ in the above result}$

Note that $h$ is a positive real number, that we let approach 0..

## Existence of Limit

For existence of limit at $x=a$

\begin{align}&\Rightarrow LHL=RHL\\ &\Rightarrow L_1=L_2\\ &\Rightarrow\lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)\end{align}

### Illustrating the concept

If $f(x)=\frac{x^2-4}{x-2}$ , then evaluate $\lim_{x\to2}f(x)$ .

L.H.L. = $\lim_{x\to2^-}\frac{x^2-4}{x-2}$ i.e. $x$ is coming closer to 2 but it is less than '2'. So, observe the situation in table below:

$x$ $2-x$ $f(x)$
1.9 0.1 3.9
1.99 0.01 3.99
1.999 0.001 3.999
$\vdots$ $\vdots$ $\vdots$
Coming closer to 2 but less than 2 Coming closer to 4 but less than 4
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