91Ó°ÊÓ

One example of a function $f$ and a value $c$ such that $\lim _{x \rightarrow c} f(x)$ happens to be equal to $f(c)$ is given as below:

$\lim _{x \rightarrow 1} 2=2$

$\lim _{x \rightarrow 1} 2=2$

Here, for $x \rightarrow 1^{-}$, the value of the function is $f\left(1^{-}\right)=2$.

Again, for $x \rightarrow 1^{*}$, the value of the function is $f\left(1^{+}\right)=2$.

One example of a function $f$ and a value $c$ such that $\lim _{x \rightarrow \mathbb{c}} f(x)$ is not equal to $f(c)$ is given as below:

$\lim _{x \rightarrow 1} x$

$\lim _{x \rightarrow 1} x$

Here, for $x \rightarrow 1^{-}$, the value of the function is $f\left(1^{-}\right)=0.999$.

Again, for $x \rightarrow 1^{+}$, the value of the function is $f\left(1^{*}\right)=1.0001$.

One example of a function $f$ and a value $c$ such that $\lim _{x \rightarrow c} f(x)$ exist but $f(c)$ does not exist is given as belowr:

$\lim _{x \rightarrow 1} \frac{1}{x-1}$

$\lim _{x \rightarrow 1} \frac{1}{x-1}$

Here, for $x \rightarrow 1^{-}$and $x \rightarrow 1^{*}$, the value of the function exists but $f(1)$ does not exists.