Question

What is a left-hand limit?

Answer 1

A limit is the value that a function (or sequence) "approaches" as the input (or index) "approaches" a certain value in mathematics.
In mathematics term, limit is usually written as –
$\mathop {\lim }\limits_{x \to c} f(x) = L$

Complete answer:
Sometimes, there is an issue that the function $f(x)$ is not defined at the given point $c$. At this time, we will have to try to find out limits around the given point $c$. So, we can either find out limits of numbers less than $c$or greater than $c$.
This concept is termed as one-sided limit. It is of two types – left-hand limit and right-hand limit.
Left Hand limit:
When we start finding the behaviour of any function around the points less than c for any function $f(x)$then the concept of left-hand limit arises.
Definition: Let’s assume a function $f(x)$ defined on any interval $(a,c)$ for every $c < b$. Then, its left-hand limit can be written as –
$\mathop {\lim }\limits_{x \to {c^ - }} f(x) = L$, for every $x$ less than $c$. It means for every number, $\varepsilon < 0$, there exists a $\delta$, such that if $c - \delta < x < c$, then $\varepsilon < |f(x) - L|$.
To define more clearly, there is an example given below –
Let’s assume a function $f(x)$ $= \dfrac{1}{x}$ and find the limit at the point $c = 0$.
Mathematically, limit function is written as –
$\mathop {\lim }\limits_{x \to 0} \dfrac{1}{x}$
When we substitute $x$ with $0$ in a function, we can see that the result will be undefined.
Now, we will try to check the behaviour of the function around the numbers less than $0$.
Let’s check the limits of the function as it approaches from the left-side to $0$.
$\ f( - 1) = \dfrac{1}{{ - 1}} = - 1, \\\ f(\dfrac{{ - 1}}{{10}}) = \dfrac{1}{{\dfrac{{ - 1}}{{10}}}} = - 10, \\\ f( - 1000) = \dfrac{1}{{ - 1000}} = - 0.001 \\\ \$
So, we can see that, limit of the function increases in the negative direction as we tend to approach the point $x = 0$.
Hence, we can conclude that, $\mathop {\lim }\limits_{x \to {0^ - }} \dfrac{1}{x} = - \infty$ .
Like this, we can find the left-hand limit of any function $f(x)$ around any point $c$.

Note:
When a function's left-hand side limit differs from its right-hand side limit, we can deduce that the function is discontinuous at the number in question.