Question

Define the left hand limit.

Answer 1

The concept of the limits and continuity is one of the most important terms to understand to do calculus. A limit is stated as a number that function reaches as the independent variables of the function reaches a given value. For example, consider a function $f\left( x \right)=12x$ , we can define this as the limit of $f\left( x \right)$ as $x$ reaches close by $2$ is $24$. We use the concept of limits to describe whether a given function is continuous or not.

Complete step-by-step solution:
As we have seen, the limit of the function $f\left( x \right)=12x$ is $24$ as $x$ reaches close to $2$.
The mathematical representation of this is as follows :
$\Rightarrow \displaystyle \lim_{x \to 2}12x=12\times 2=24$ .
Limits are very useful in determining whether a function is continuous or not. This is where both the limits namely left hand limit and right hand limit come into picture.
A left hand limit is defined as the limit of a function as it approaches from the left hand side.
Let us our function $f\left( x \right)=12x$ . Since we are coming from the left hand side of $2$. It is important to specify that. We specify it by adding a negative symbol (-) on top of $2$ like this ${{2}^{-}}$ .
Our left hand limit will be the following :
$\Rightarrow \displaystyle \lim_{x \to {{2}^{-}}}12x=12\times 2=24$
A right hand limit is defined as the limit of a function as it approaches from the right hand side.
Let us our function $f\left( x \right)=12x$ . Since we are coming from the right hand side of $2$. It is important to specify that. We specify it by adding an additional symbol (+) on top of $2$ like this ${{2}^{+}}$ .
Our right hand limit will be the following :
$\Rightarrow \displaystyle \lim_{x \to {{2}^{+}}}12x=12\times 2=24$
The left hand limit and the right hand limit seem to be equal .
Since they are both equal, we can say that our function $f\left( x \right)=12x$ is continuous. If both of them are not equal , then a particular function is not continuous.

Note: Limits is a very important part of calculus. Limits are used to determine whether a function is continuous for some particular value of $x$. It is used to determine whether a function is differentiable at some particular value of $x$. It is also used in definite integration. We should all of it’s theorems and other basic formulae. We should also learn the expansions of a few trigonometric and exponential functions.