FANDOM


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
Community content is available under CC-BY-SA unless otherwise noted.