Left-Hand Limits and Right-Hand Limits - Practice Questions & MCQ

Left-Hand Limits and Right-Hand Limits

Continuing from the previous concept, we can approach the input of a function from either side of a value the left or the right.
we had our function as
$f(x)=\frac{(x+1)(x-7))}{(x-7)},x\ne 7$ which becomes equivalent to the function

$f(x)=x+1,x\ne 7$

now, let us observe the values of $f(x)$ nearby $x=7$.

Approaching 7 "from the left" means that the values of input are just less than 7. And if for such values of $x$, the values of $f(x)$ are close to $L$, then $L$ is called the left-hand limit of a function at $x=7$. For this function, 8 is the left-hand limit of the function $f(x)$ at $x=7$.

Approaching 7 "from the right" means that the values of input are just larger than 7. And if for such values of $x$, the values of $f(x)$ are close to $R$, then $R$ is called the right-hand limit of a function at $x=7$. For this function, 8 is the right-hand limit of the function $f(x)$ at $x=7$.

To indicate the left-hand limit, we write $\underset{x\to 7^{-}}{lim}f(x)=8.7$ - indicates the values that are less than 7 and are infinitesimally close to 7

To indicate the right-hand limit, we write $\underset{x\to 7^{+}}{lim}f(x)={8.7}^{+}$indicates the values that are greater than 7 and are infinitesimally close to 7.

The left-hand and right-hand limits are the same for this function. The point can be defined)
The left-hand limit of a function $\mathrm{f}(\mathrm{x})$ as x approaches a from the left is denoted by $\underset{x\to a^{-}}{lim}f(x)=LHL$
The right-hand limit of a function $\mathrm{f}(\mathrm{x})$ as x approaches a from the right is denoted by $\underset{x\to a^{+}}{lim}f(x)=RHL$
Now consider a function, $f(x)=\frac{|x|}{x}$
Let's check the behavior of $f(x)$ in the neighborhood of $x=0$

$\mathrm{LHL}=\underset{\mathrm{x}\to 0^{-}}{lim}\frac{|\mathrm{x}|}{\mathrm{x}}$
As x is just less than 0, we can replace it by $(0-h)$, where $h$ is positive and very close to 0

$=\underset{\mathrm{h}\to 0^{+}}{lim}\frac{|0-\mathrm{h}|}{0-\mathrm{h}}$
So, we have

$\begin{array}{rl}& =\underset{\mathrm{h}\to 0^{+}}{lim}\frac{|-\mathrm{h}|}{-\mathrm{h}}\\ & =\underset{\mathrm{h}\to 0^{+}}{lim}\frac{\mathrm{h}}{-\mathrm{h}}\\ & =-1\end{array}$
$\begin{array}{rl}\mathrm{RHL}& =\underset{\mathrm{x}\to 0^{+}}{lim}\frac{|\mathrm{x}|}{\mathrm{x}}\\ & =\underset{\mathrm{h}\to 0^{+}}{lim}\frac{|0+\mathrm{h}|}{0+\mathrm{h}}\\ & =\underset{\mathrm{h}\to 0^{+}}{lim}\frac{\mathrm{h}}{\mathrm{h}}=1\end{array}$
Here, we have RHL $\ne \mathrm{LHL}$

Existence of a limit of a function

From the above example, we can define the existence of a limit
The limit of a function $f(x)$ at $x=a$ exists if $\underset{x\to a^{-}}{lim}f(x)=\underset{x\to a^{+}}{lim}f(x)$ or $\underset{h\to 0^{+}}{lim}f(a-h)=\underset{h\to 0^{+}}{lim}f(a+h)$.
i.e., $LHL=RHL$ at $x=a$

Also, notice that the limit of a function can exist even when $f(x)$ is not defined at $x=a$.