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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 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}

Light Hand Limit

If f(x) approaches a form left, then L_2 is called as the left hand limit of f(x) . 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 an infinitely small positive number approaching to 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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