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
$f(x)=\frac{(x+1)(x-7))}{(x-7)}, x \neq 7$ which becomes equivalent to the function

$
f(x)=x+1, x \neq 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 $\lim _{x \rightarrow 7^{-}} f(x)=8.7$ - indicates the values that are less than 7 and are infinitesimally close to 7

To indicate the right-hand limit, we write $\lim _{x \rightarrow 7^{+}} f(x)=8.7^{+}$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 $\lim _{x \rightarrow a^{-}} f(x)=L H L$
The right-hand limit of a function $\mathrm{f}(\mathrm{x})$ as x approaches a from the right is denoted by $\lim _{x \rightarrow a^{+}} f(x)=R H L$
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}=\lim _{\mathrm{x} \rightarrow 0^{-}} \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

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

$
\begin{aligned}
& =\lim _{\mathrm{h} \rightarrow 0^{+}} \frac{|-\mathrm{h}|}{-\mathrm{h}} \\
& =\lim _{\mathrm{h} \rightarrow 0^{+}} \frac{\mathrm{h}}{-\mathrm{h}} \\
& =-1
\end{aligned}
$
$
\begin{aligned}
\mathrm{RHL} & =\lim _{\mathrm{x} \rightarrow 0^{+}} \frac{|\mathrm{x}|}{\mathrm{x}} \\
& =\lim _{\mathrm{h} \rightarrow 0^{+}} \frac{|0+\mathrm{h}|}{0+\mathrm{h}} \\
& =\lim _{\mathrm{h} \rightarrow 0^{+}} \frac{\mathrm{h}}{\mathrm{~h}}=1
\end{aligned}
$
Here, we have RHL $\neq \mathrm{LHL}$

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 $\lim _{x \rightarrow a^{-}} f(x)=\lim _{x \rightarrow a^{+}} f(x)$ or $\lim _{h \rightarrow 0^{+}} f(a-h)=\lim _{h \rightarrow 0^{+}} f(a+h)$.
i.e., $L H L=R H L$ at $x=a$

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